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September 22 – York Minster
Barley Hall, Yorvik Dig and Clifford's Tower
Hello from 1066 – Medieval Mosaic
At 10 we left the Pinfold Cottage in Fulford and drove into York for the day. We soon realised that we should have just left the car behind and gone in by bus as it was £8.80 for four hours to park. Still, it had been a bit of an ordeal to find the Nunnery Lane parking and but the car parks all seem to be a similar price. We then realised that it was quite a walk into town, not to worry. Our target for today was the York Minster. It costs £10 each to go in but you do get free entry for a year afterwards. We got there just before the free tour starts so we happily joined the tour as you learn so much more that way. There was a good size group of about 30 and Mike Bradshaw was our guide. I took notes as we went as otherwise, I forget so easily. So much information in a short space of time.
The Cathedral and Metropolitical Church of Saint Peter in York, commonly known as York Minster and is one of the largest of its kind in Northern Europe. There have been several churches on this site since 627, the second was built after a fire in 741. There was a series of Benedictine archbishops, including Saint Oswald of Worcester, Wulfstan and Ealdred, who travelled to Westminster to crown William in 1066. Ealdred died in 1069 and was buried in the church. After the cathedral was damaged during the harrying of the North, Thomas of Bayeux became the first Norman Archbishop and he organised repairs. It was damaged and repaired several times until in 1215 Walter de Gray, a favourite of King John, who was present at the signing of the Magna Carta, was made archbishop and ordered the construction of a Gothic structure. The building, in Tadcaster limestone, took place around the original structure doing one area at a time and continued until the 15th century. Walter has an impressive tomb inside the minster. but never lived to see the final cathedral. Walter also built Bishopthorpe Palace, where he lived. The current Archbishop also lives there. The Wikipedia account of York Minster is well worth a read. There was also a Roman headquarters on the site before the churches, where the administration of the legion and religious ceremonies took place. Constantine was proclaimed Emperor of the Roman Empire here. He tolerated Christianity then became Christian and after his death, it became the religion of the Empire in 306AD.
There are 128 stained glass windows in the Minster and many are Medieval Glass. The West window of 1330 is called 'the heart of Yorkshire' and depicts Mary and Jesus at the top, below are bible stories, blow that are the apostles and below that the Archibishops. Below this is a modern sculpture of headless characters spelling out 'Christ is here' in semaphore. There was a fire in 1840 which took at the roof at the West end as it was made of wood. The carvings across the roof were lost but drawings had been made so that they could be replaced as original except one. This one had previously shown Mary breastfeeding. In Victorian times when the rebuilding took place, this was unacceptable so it now shows Mary bottle feeding. One particularly huge window in the North transept is grisaille glass. This window is made from 100,000 pieces of nearly all grey glass except for one place which is a more recent mend. The older glass in this window dates from 1100 and it is 16 ft high in geometric designs.
The Chapterhouse is where the Dean and the College of 44 Canons meet to discuss and plan, the canons represent all the geographical areas that the cathedral covers. The Chapterhouse has a suspended ceiling instead of having a centre supporting pillar which was built between 1260 and 1280. The first female Church of England Bishop, Libby Lane, was consecrated here in 2015. Oliver Cromwell banned chapter houses and it was nearly sold at that time.
There is only one royal tomb in the cathedral which is Prince William of Hatfield, son of Edward III, who died in 1335. Although depicted as a teenager on the tomb he actually died aged only 5 months.
The tower is 60 metres high and has 275 steps going up to it, making it the highest place in York. The roof with the bells weighed 16,000 tons. It collapsed in 1407 due to the soft soil beneath its foundations. The bells were not replaced as the weight was too much for the structure. The tower was in danger of collapsing again in the 1960s before major structural work was carried out to reinforce its foundations – the space created during these works now houses the cathedral's Undercroft attraction which is really interesting. The tower is big enough to hold the Statue of Liberty.
On 9 July 1984, a fire believed to have been caused by a lightning strike destroyed the roof in the south transept, and around £2.5 million was spent on repairs. The restoration work was completed in 1988 and included new roof bosses to designs which had won a competition organised by BBC Television's Blue Peter programme. In 1986 Blue Peter ran a competition to find six boss designs with a modern theme for the Minster's vaulted ceiling. Six designs were chosen from 34,000 entries including Neil Armstrong, Save the Whales and the Rasing of the Mary Rose.
The Kings' Screen, also known as the Quire Screen, is one of the most famous parts of York Minster. Unusually, the screen is asymmetrical, featuring 15 carved statues of kings of England, which means the doorway leading to the Quire is off centre. The screen was designed around 1420 and features English kings ranging from William the Conqueror to Henry VI. It's believed it was originally intended to feature 14 statues up to Henry V, but a hasty revision was needed when his reign was unexpectedly cut short in 1422 before the screen was finished, and it became necessary to include Henry VI.
The quire had a fire in 1829 and all the woodwork was lost.
The Great East window, which is the size of a tennis court and is the single largest expanse of medieval stained glass in the country, is currently undergoing a major restoration costing £20 million over 5 years. It was created between 1405 and 1408 by master glazier John Thornton and shows the start and end of all things. The cleaning and restoration will be completed by Easter 2018. The plain glass part at the top is the part still being worked on.
After the tour was over Mike Bradshaw came over to us and asked where we were from. When we said Geraldine, NZ, he exclaimed 'that's where the Giant Jersey is'. Small world!! He had even purchased our USB about the Medieval Mosaic. We talked to him and his colleagues for a while and they gave us contact details of the exhibitions and collections manager for York Minster so that we will be able to write to her about the possibility of holding an exhibition there.
After this, we visited Barley Hall and the Yorvik Dig as both of those we gained free entry with the Yorvik passport which we got yesterday at the Yorvik Centre. We also have free entry to all these three and two other exhibitions for a full year. Barley Hall is a reconstructed medieval townhouse in the city of York, England. Originally built around 1360 by the monks of Nostell Priory, it was later extended in the 15th century. It was very interesting and really well done allowing visitors to touch and sit and play with objects. The Medieval dining room was particularly interesting. It showed a peacock as the centrepiece for the meal. Apparently, the cooks would carefully remove the feathers and skin from the bird, the roast the bird before replacing the skin and the feathers to serve it as a complete peacock. Fantastic but highly dangerous from a health point of view.
The Dig is mostly a children's exhibition where bones and artefacts found during the dig before the building of the supermarket complex and is one of the few places in the country that specialises in interactive archaeology. We found it very interesting though as their displays were excellent especially the part about Rowntree's. Rowntree was an English confectionery business based on Quaker principles in York. Rowntree developed the Kit Kat (introduced in 1935), Aero (introduced in 1935), Fruit Pastilles (introduced in 1881), Smarties (introduced in 1937) brands, and the Rolo and Quality Street brands when it merged with Mackintoshes in 1969.
On returning to the car after moving it to the Castle car park we walked up the English Heritage run, Clifford's Tower or York Castle. York Castle in the city of York, England, is a fortified complex comprising, over the last nine centuries, a sequence of castles, prisons, law courts and other buildings on the south side of the River Foss. It was built originally on the orders of William I to dominate the former Viking city of York.
We spent a wonderful 6 hours in York exploring, however, we barely scratched the surface as we didn't visit the museum, railway museum, art gallery or either of the exhibitions situated on the city walls. So we will definitely return, quite possibly our favourite city yet.
We ended the day by driving to Bilborough Travel Lodge where we spent a comfortable night after having our first ever meal at Little Chef.
All Saints, Pavement, York
Royal Coat of Arms of William III and Mary II (1689 - 1694) - All Saints, Pavement, York
Alabaster reredos with Christ in the centre flanked by the apostles and the evangelists - All Saints, Pavement, York
The font - All Saints, Pavement
Remembrance window by Helen Whittaker of Barley Studios - All Saints, Pavement
Stained glass window. Nativity scene with censing angels by Kempe - All Saints, Pavement
Hannah and Samuel, Mary, infant Jesus and St Elizabeth - Stained Glass by Kempe, 1891 - All Saints, Pavement
The pulpit (1634), inscribed with the words "The foolishness of preaching" - All Saints, Pavement
York Minster - England's largest medieval cathedral
The Great West Door with the effigy of St Peter, York Minster
Gilded ceiling bosses - York Minster
Octagonal Chapter House - York Minster
Five Sisters' Window (c.1260) 100,000 pieces of grey and green 'grisaille' glass measuring 53 feet
West Window contains a heart-shaped design, known as 'The Heart of Yorkshire' - York Minster
The Bishop's Cathedra - York Minster
400-year-old 4 quarter chime clock by Henry Hindley (1701–1771) located in the North Transept of York Minster. The two knights strike the quarter hours by striking each other
The Organ - York Minster
The choir stalls and organ - York Minster
The rose window - York Minster
The tomb of William Markham - York Minster
Oliphant presented to York Minster by the Viking nobleman Ulf in 1030
Mid 13th century ring found in the tomb of Archbishop Walter de Gray at York Minster during a 1960's restoration
Reliquary of St William of York, 1148 - York Minster
Geneva Bible 1607 - York Minster
Kings Book of York Heroes, presented to York Minster in 1920 by the Duke of York
The York Gospels, 1020 - York Minster
The Kings Screen, York Minster. The choir sceen contains sculptures of the Kings of England from William the Conqueror to Henry VI., with stone and gilded canopies set against a red background
The Chapter House of York Minster was built around 1260. In 1297 it was used for the Parliament of King Edward I. of England. Dating from 1270, sculptured heads above the canopies represent some of the finest Gothic sculpture in the country. With no 2 heads the same, there are over 250 characters around the Chapter House
Astronomical Clock - York Minster
Memorial to Prince William of Hatfield, second son of Edward III - York Minster
Sir George Savile, 8th Baronet, (1726 - 1784) - York Minster
Sir Henry (1624) & Lady Ursula Bellasis by Nicholas Stone 1616 - York Minster
Henry Swinburne (1624) - York Minster
Ornate gilded ceiling - York Minster
Gilded Gates - York Minster
Archbishop Toby Mathew (1628). Designed by Sydney Smirke - York Minster
Dossal made in 1972 by the Minster Broderers. Design by Joan Freeman - York Minster
Sir William (1623) & Lady Catherine Ingram (1670) - York Minster
Wall monument to Lionel Ingram (1628) died aged 6. - York Minster
Wall monument to Anne Benet (1602) by Nicholas Stone - York Minster
William Wentworth, 2nd Earl of Strafford (d. 1695) & Lady Honoria (d. 1685) - York Minster
Wall memorial to Edmund Bunny (1618) - York Minster
Wall monument to Archbishop John Piers (1594) - York Minster
Sir William Gee (1611) & 2 of his wives: Thomasine (1559) & Mary (1649) - York Minster
Archbishop Walter de Gray (1215-1255) 13th century coffin of Purbeck marble - York Minster
Gilded bosses and rose window - York Minster
Archbishop John Dolben (1686) by Grinling Gibbons - York Minster
Wall monument to Nicholas (1617) & William (1577) Wanton - York Minster
Archbishop Matthew Hutton (1606) - York Minster
Left : Christopher Edward Thomas Oldfield (1850) by Mathew Skelton. Right : Canon William Mason (1797) & Canon William Dixon (1854) Inlay of coloured marbles, surface of bronze, brass, enamels and semi-precious stones; alabaster figures of the Good Shepherd and two angles. By Francis Alfred Skidmore.
York Castle Museum
Minera, Goddess of Wisdom by John Wolstenholme, 1801 - High Petergate, York
The Victorian arched roof of York Railway Station
Gillian in Clifford's Tower, York Castle
Clifford's Tower, the keep of York Castle
Looking down inside Clifford's Tower, York Castle
Gillian takes the steep steps up to Clifford's Tower, York Castle
A model of York Castle
View of York from the top of the tower
View of York Minster from the top of the tower
A visit to Barley Hall, York
Viking Kitchen - Barley Hall
Feast for a King - Barley Hall, York
Swan on the menu at Barley Hall, York
Costumes on display at Barley Hall, York
Teaching aid made of horn - Barley hall, York
A scene from the Westminster Tournemount Roll, 1511 - Barley Hall, York
Medieval Horn Window - Barley Hall, York
Fragment of St. William's Shrine - Barley Hall, York
Medieval Vovelle wheel - Barley Hall, York
15th century manuscript. The Archpriest of Talavera by Alonso Martínez de Toledo. Barley Hall, York
15th century spinning wheel - Barley Hall, York
The DIG. The DIG is a hands-on archaeological adventure discovering artefacts from 2000 years of York's history
A skeleton at The DIG. A demonstration of archaeological procedures
A Holographic presentation of archaeology at The DIG, York.
Copyright [year] M.A&G.S.Linton. All rights reserved.
Website by Redpost Media


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Ajnianwala is a town of Sheikhupura District in Punjab, Pakistan. It is  from Sheikhupura on Hafizabad Road.

History
Ajinanwala dates back at least 500 years. Before the partition of Indian subcontinent, the town had a considerable Sikh majority, who upon partition migrated to India. Gurdwara Ajnianwala is still intact but was allocated to migrants from Indian Punjab in 1947 which is now non-functional. This place was built by Surjan Das Khatri and the "Surjan Dasia" sect draws its name from him. Samadhs outside the Gurdwara belong to Surjan Das, Athal Das and two of their disciples. The Granth compiled by Surjan Das is known, as "Ajat Granth". Fare of Smadh used to be held in Chait before partition.

Like many other classic Punjabi towns, a fair (maila) is held in Chait (March 16–18) too where the traditional game of Punjab, kabbadi, is played. This fair is associated with the saint Baba Sadiq Shah Wali, whose tomb is also in Ajnianwala.

People of Ajnianwala

Ajnianwala is multicultural and multiethnic town. The majority of population is local Punjabi-speaking while around 40% people are descendants of migrants from Indian Punjab. Major casts of Ajnianwala include Guru Jatt, Goraya jatt, Walana Jutt, Jutt spall, Khara, Janjua, Shaikh, Araien, Syed, Malik and Rajputs or Ranas, Mayo, Mughal, Awan, Khokher, Ansari.Sial.

The town is a commercial hub for adjacent rural areas. Two canals, the Qadirabad-Balloki link and Upper Gogera Canals, flow beside the town and irrigate the land.

Education

Ajnianwala serves as education centre of the area where two higher secondary schools and colleges for boys and girls besides two middle schools for boys and girls operate. A number of private schools are also in town.

 Seven Star Model School, Ajnianwala
 The Talent College of science, Ajnianwala.
 Educants Science Academy Ajnianwala, Sheikhupura
 Decent public school, Ajninawala.
 Misaali Grammar school, Ajnianwala.
 Hayat Cambridge school, Ajnianwala.
 The HOPE school system, Ajnianwala.
 M.A Azad Public Model School Ajnianwala.
 EFA school, Ajnianwala.
 Allied school, ajnianwala.
 Dar-e-Arqam school, ajnianwal.
 Wisdom house tuition centre, Ajnianwala.
 Rana Science Academy

Shops
Mobile shops:

Alfazal mobile shop

Faisal mobile shop

New Ali Mobile shop

Afzal mobile shop

Husnain traders

Sahi mobile shop

all shops are located on hafizabad-sheikhupura Road near United Bank ajnianwala

Super Stores:

Following located on khanqa dogran Road near United bank

Alburaq Super Store almashoor sheikhan di hatti

AlFiaz super store

Sheraz general store

Sarfraz super store

Faisal karyana store

Naveed General store

Following located on main road

Faizan karyana store

Sheikh jameel karyana store

Toheed karyana store

Bilal super store

Health

The main health centre is Basic Health Unit Ajnianwala, while there are number of private clinics providing health facilities to the general public.

Public offices

There is a police Chowki in the town with pre-partition history. Union Council Ajnianwala serves as a local government institution while the post office has been operating since the British era.

References

Populated places in Sheikhupura District

redpajama_set_name: RedPajamaWikipedia


text:
DeMarcus Ware, Broncos agree on three-year, $30 million deal
160 posts	• Page 5 of 8 • 1, 2, 3, 4, 5, 6, 7, 8
Re: Article Discussion: DeMarcus Ware signs with Broncos as John
by GoBroncos24 on March 12th, 2014, 4:29 pm #3328404
300Paul wrote:
ibcleveland wrote:
ibcleveland wrote: Too bad they didn't do this a year ago. If they had, they just might be the Super Bowl Champions now instead of a team that got the worst pummeling in SB history.
That wasn't the worst loss in SB history. Haven't been watching the game long I take it?
Yup. Those were pathetic but this b-slapping by Seattle I think was worse. The teams that lost to Washington and SF were over achievers who, let's be honest, made it to the Big Game on the back of one John Elway who managed to drag the Broncos to the Super Bowl in spite of a relatively mediocre supporting cast and the hugely unimaginative, plodding coaching of Dan Reeves. This year, the Broncos had the best offense in NFL history and what...? The first snap from center sails nervously over Manning's head for a safety. And then it got ugly. And then it got uglier. And then uglier still. If beauty is in the eye of the beholder then so is ugly. And this was the ugliest yet.
You thought what happened against Seattle was WORSE than what San Fran did? You gotta be kidding me.
It's awfully close, and I might agree it's worse, given how great we were the rest of the year.
I mean, the idea of getting blown out seemed IMPOSSIBLE to me given how we looked against San Diego and New England.
Of course, back in 1989, I'm pretty sure I was also very confident going in.
Just a bizarre Super Bowl...still don't get what exactly happened...even with all of our injuries...even with the return of Harvin...if we played again, we would likely lose, but likely by closer to 7-10 points, not 35.
GoBroncos24
Mike C
Re: Article Discussion: DeMarcus Ware, Broncos agree on three-ye
by retro-grouch on March 12th, 2014, 4:30 pm #3328409
GoBroncos24 wrote: It just depends....I will agree it's unlikely any rookies will come in and start right away, unless we go o-line or maybe WR.
Fox likes to set rookies up in competition with veterans but the right rookie at WR or MLB could actually win one of those bouts by mid season.
It would take an exceptionally smart rookie but its not out of the question. On offense, once Manning figures out that a young player is motivated and capable of learning he will hammer them in practice.
At MLB their free agents have been so weak that you have to give the rookie a shot.
Last edited by retro-grouch on March 12th, 2014, 4:33 pm, edited 1 time in total.
retro-grouch
Mike J
Joined: April 22nd, 2010, 7:41 am
by 300Paul on March 12th, 2014, 4:31 pm #3328410
GoBroncos24 wrote:
kenplant2007 wrote:
BroncfanOR wrote: The question is now where do they go in the draft?
Let's assume they are sliding Franklin to LG and Clark to RT they still need some depth.
They lack a MLB unless they believe Irving is going to step up this year but I'm not seeing that as happening. They still need a young WR who hopefully can return as well and possibly another CB.
If Denver is TRULY going all in as it appears I could see them trading up, per the rumors for Mosley to play the MLB. Of course that will come at a HUGE cost which they may not be willing to pay.
That being the case do they really take best player available? Do they target one of the other MLB's like Skov in the second round?
The draft just got more interesting for the Broncos in my opinion.
why are you concerned about the draft? all their picks take 3 years to develop...and this team will be over by then
Except that Sly Williams and Montee Ball will both be starting this season...in their 2nd year.
And look for Kayvon Webster to have an expanded role too.
And it only took Trevathan one season to get a big role with the team.
It just depends....I will agree it's unlikely any rookies will come in and start right away, unless we go o-line or maybe WR.
It's good to have a team that is good enough where rooks don't have to play right away, it means the team is good, some don't see it that way, it's a 50/50 argument I guess.
Yeah, it's just hard to think of any position that a rookie could immediately start, maybe besides guard or center, if they didn't shift Franklin/Clark?
WR maybe...but that's a tough gig for a rookie with Peyton Manning, at least early in the season.
I'm sure we'll need a solid rookie CB for the nickel/dime though.
Agreed. And many people really seem to forget, all rookies hit that wall.
I much rather be in the Broncos position then like a team like, say the Browns, where if your rookie who HAS to play, plays like a rookie, never mind how good he will be in the future, then that teams season is sunk, AGAIN. I like the set up that Broncos have, hard not to.
300Paul
Paul Y
Every once in a blue moon you get a rookie who plays like a veteran.
I'm looking back at it from the eyes of a child, which I was when S.F. did their thing, but S.F. was a complete juggernaut then, I still feel the Broncos gave it away to Seattle, but that S.F. game, they were outclassed, and that was obvious from the jump, it was straight murder.
Yup, one of the great teams of all-time in SF that year...just a machine.
by BroncfanOR on March 12th, 2014, 5:03 pm #3328429
Really dude? It's time to put your big boy pants on and stop acting like a child.
Your post is ridiculous.
BroncfanOR
Dave W
Dave's recent comments
Article Discussion: DeMarcus Ware signs with Broncos as John
by gwats on March 12th, 2014, 5:05 pm #3328434
Been here 35 years.... knew the City had a 'terminal case' of 'SOS' ( Stuck on Stupid), but until today, I didn't know John Elway, a man I've admired for a very long time, was suffering from this affliction also. Go Broncos..... and take the Nuggets WITH YOU! Seriously, Andre Ware? keep on shopping for other team's rejects...
G. W
G.'s recent comments
by Mrzuki on March 12th, 2014, 5:08 pm #3328437
gwats wrote: Been here 35 years.... knew the City had a 'terminal case' of 'SOS' ( Stuck on Stupid), but until today, I didn't know John Elway, a man I've admired for a very long time, was suffering from this affliction also. Go Broncos..... and take the Nuggets WITH YOU! Seriously, Andre Ware? keep on shopping for other team's rejects...
I hope I got here first ---- "Andre Ware" are you kidding?
Keep on admiring John Elway, and get some help.
It's Demarcus Ware.
Mrzuki
Dan e
Dan's recent comments
Joined: September 21st, 2008, 7:23 am
You knew the city was stuck on stupid for 35 years and you stayed??? HAHAHAHAAHAAA! If Ware is the only thing you are pi$$ed about, Elway is still doing kinda o.k.
Agreed on the Nuggets though, yuck.
Speaking of terminal cases… Andre Ware was the 1989 Heisman winner.
Anyway, Jerry Jones screwing up his cap and being forced to cut his best player is not exactly rejection. Plus, you being here 35 years can't have helped the average I.Q.
Last edited by retro-grouch on March 12th, 2014, 5:18 pm, edited 2 times in total.
Mrzuki wrote:
You know what? You're right and I was wrong in my reply to gwats, except for the Nuggets thing.
I too would be pissed if Elway had given Andre Ware a contract. That would be quite the reclamation project, and I wold just assume he would suck at getting to the Q.B.
DEMARCUS WARE probably has more sacks then Andre had completions, but, whatever, ya know?
(edit)***Just to kick someone when they are obviously down, ANDRE Ware had 83 completions in his career, DEMARCUS Ware has 117 sacks...and that was just an assumption, at first.
Classic gwat, just classic.
Last edited by 300Paul on March 12th, 2014, 5:22 pm, edited 2 times in total.
Yup, Andre Ware. It's amazing, because he's like 50 now since he last played in the 90's. Seems like a weird move.
We should've gone for DeMarcus Ware, who could've been huge.
You know what? You're right and I was wrong in my reply to gwats, except for the Nuggets thing. I too would be pissed if Elway had given Andre Ware a contract. That would be quite the reclamation project, and I wold just assume he would suck at getting to the Q.B.
I'm picturing Andre, I mean Demarcus coming off the right side and Von coming off the left. We will have some combination of Pot Roast, Kevin Vickerson, Wolfe, Sly, Malik, Queanterous (sp). Add to that Talib and Ward, who not only cover but they hit. We now have the speed, ability and most importantly the attitude on defense to compete with the NFC West, who we have to play next year in the regular season.
And if that doesn't work out, maybe Andre Ware can look at backing up Dysert.
Ding ding ding. We have a winner for worst post of the day.
Tell him what he's won............
by dynamicdave on March 12th, 2014, 5:39 pm #3328459
Damn, guys. I was hoping to pile on, but you guys already dished it out pretty good. Some people like to gripe, no matter the reason or even if the facts are correct. Gnads, er, I mean, qwats, needs to get a hobby.
dynamicdave
David T
David's recent comments
Revis has agreed to a one year deal with New England.
Reported to be 12mil
BroncfanOR wrote:
An Andre Ward t-shirt? I hear those are free these days.
by HaleAndHearty on March 12th, 2014, 5:46 pm #3328466
I've never really been against piling on.
I thought for a split second y'all were taking about Charlie Ward. I'd just awakened from a well-deserved nap, where I was dreaming in orange Technocolor about Ware and Miller giving a squeegee sloberknocker to Phyllis Rivers ... and then I had a horrid tebow flashback, thought maybe the team signed Ward to back up tebow or something ...
I got really ticked. Then I regained my senses, and had a constructive thought. I hope gwats is a season ticket holder. I mean, we all have to contribute somehow, right?
Last edited by HaleAndHearty on March 12th, 2014, 5:55 pm, edited 1 time in total.
HaleAndHearty
mr j
mr's recent comments
BroncfanOR wrote: Revis has agreed to a one year deal with New England.
See how this is shaping up Oregano?
In the playoffs next season, Revis pretty much shuts down D. Thomas, until late in the 4th, at a critical juncture, Welker takes out Revis. It's the WORSTEST THING Dark Billy has ever seen. Immediately following, D. Thomas scores a TD on a Manning throw to put the game away.
AND BEFORE THAT Bubba Caldwell goes for about 225 and 4 TDs.
It's all good man.
The System by Mr. B

redpajama_set_name: RedPajamaCommonCrawl


Another great experience with Darwin.
Great customer and really helpful when I had a server problem. The OverDelivery is still coming in today. Thank you for your repeat custom, Darwin.
Great customer; thank you once again.
Great buyer; thank you, Darwin.


From the fabric to the finished garment, all in one company, which follows every step of production. A company made up of people, knowledge, values, which are transmitted daily in every product object.
Beste offers the opportunity to visit the company with a guided tour of the entire production chain from the design stage to all processes, where each department and machinery is identified by a special sign with the name and QR Code for further study, until the realization of finished garments. At the end of the tour it will be possible to consult the TESSUTEKA, an archive with over twenty-five years of fabrics produced.
Company and production cycle tour.
Workshop: development of a virtual fashion collection.


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Leon Albert Henkin (April 19, 1921, Brooklyn, New York - November 1, 2006, Oakland, California) was an American logician, whose works played a strong role in the development of logic, particularly in the theory of types. He was an active scholar at the University of California, Berkeley, where he made great contributions as a researcher, teacher, as well as in administrative positions. At this university he directed, together with Alfred Tarski, the Group in Logic and the Methodology of Science, from which many important logicians and philosophers emerged. He had a strong sense of social commitment and was a passionate defensor of his pacifist and progressive ideas. He took part in many social projects aimed at teaching mathematics, as well as projects aimed at supporting women's and minority groups to pursue careers in mathematics and related fields. A lover of dance and literature, he appreciated life in all its facets: art, culture, science and, above all, the warmth of human relations. He is remembered by his students for his great kindness, as well as for his academic and teaching excellence.

Henkin is mainly known for his completeness proofs of diverse formal systems, such as type theory and first-order logic (the completeness of the latter, in its weak version, had been proven by Kurt Gödel in 1929). To prove the completeness of Type Theory, Henkin introduces new semantics, based on certain structures, called general models (also known as Henkin models). The change of semantics that he proposed permits to provide a complete deductive calculus for Type Theory and for Second-Order Logic, amongst other logics. Henkin methods have aided to prove various model theory results, both in classical and non-classical logics. Besides logic, the other branch on which his investigations were centered was algebra; he specialized in cylindric algebras, in which he worked together with A. Tarski and D. Monk. As for the philosophy of mathematics, although the works in which he explicitly approaches it are scarce, he can be considered to have a nominalist position.

Life

Childhood and first youth
Leon Albert Henkin was born on April 19, 1921, in Brooklyn, New York, to a Jewish family that had emigrated from Russia a generation earlier. The first of the family to emigrate was Abraham Henkin, the eldest of the brothers of Leon's father. According to Leon, his father had been extremely proud of him since he was just a boy. His high expectations were evident in the name he gave him: he chose to name his son Albert after a series of articles on Einstein's theory of relativity that the New York Times published shortly before Henkin's birth. His family was sympathetic with pacifist and progressive ideas, and although he was not religious, he had deeply rooted Jewish traditions. Leon grew up surrounded by tight family ties; he was very close to his cousins, with whom he lived during his childhood in Brooklyn.

Henkin studied primarily in New York City public schools; he attended Lincoln High School, where he graduated at age 16 to enter Columbia University. Both in college and high school he was a member of the chess teams; he always preferred games that involved rational thinking to games of chance. In the years of his high school education, Henkin considered becoming a math teacher and also came to desire to become a writer (as he later expressed in a personal letter). Although he dedicated himself to university academic life, he never abandoned his interest in teaching elementary mathematics, to which he later actively contributed.

The first university studies
In 1937 Leon entered Columbia University as a mathematics student. It was during his time at this institution that he developed an interest in logic, which would determine the course of his academic career. His first contact with logic was through B. Russell's book, "Mysticism and Mathematics", which drew his interest during a visit to the library. This interest was increased and cultivated by some courses. Although the mathematics department of the University did not offer courses in Logic (these were offered by the Philosophy department), Leon was one of the few mathematics students interested in that discipline and he decided to attend them. In the fall of 1938, in his second year as a Columbia University student, he participated in a first course in Logic taught by Ernest Nagel, who had contributed to the creation of the Association of Symbolic Logic two years earlier. This course brought him closer to Russell's book "Principles of Mathematics", where he first encountered the axiom of choice; Russell's presentation made a strong impression on him and led him to explore the Principia Mathematica that Russell wrote with Whitehead a few years later. He was struck by the general ideas of Type Theory and by the mysterious axiom of reducibility. Both the axiom of choice and Type Theory later played an important role in his doctoral dissertation.

The following year, in the fall semester of 1939, Henkin took a second course of Logic with Nagel, in which formal systems of propositional logic and First-Order Logic were addressed. These constituted his first experience with the mathematical treatment of deductive systems. The course did not go into metalogical results that established a relationship between the semantics and syntactics, and the issue of completeness was not addressed at all. However, Nagel proposed to Henkin as an independent project the reading of the proof of the completeness of propositional logic given by Quine, which had appeared a few months before in the Journal of Symbolic Logic. This reading was highly significant for Henkin, not so much because of the content itself, but because with it he discovered that he could understand the research on logic and mathematics that was taking place at the time. According to Henkin, although he managed to follow Quine's demonstration, he did not manage to capture the idea of the proof: "I simply noted that the aim of the paper was to show that every tautology had a formal proof in the system of axioms presented, and I expended my utmost effort to check Quine's reasoning that this was so, without ever reflecting on why author and reader were making this effort. This strictly limited objective also kept me from wondering how the author thought of putting the steps of the proof together; the result was that I failed to get 'the idea of the proof', the essential ingredient needed for discovery."

Just before Henkin began his second year at Columbia, World War II broke out. This had several repercussions on his life. One of them had a positive effect on his education. Days before the war broke out, the Polish mathematician and logician Alfred Tarski had come to Harvard, at Quine's invitation, to give a series of lectures on logic. With the invasion of Poland by Germany, Tarski found it impossible to return to Poland and he had to remain in the United States. Tarski visited several cities giving lectures on logic. One of these lectures was at Columbia, and Henkin, like the rest of the logic students, attended it with great enthusiasm. In it Tarski spoke of Gödel's work on undecidable propositions in Type Theory and on the existence of decision algorithms for formal systems, a subject that Henkin found extremely stimulating.

In his last year at Columbia, in 1941, Professor F. J. Murray, knowing that Henkin was a mathematics student interested in Logic, suggested that they review together the monograph by Gödel recently published at Princeton on the consistency of the axiom of choice with the generalized continuum hypothesis. Although the meetings they had to discuss it were scarce and Leon ended up revising this monograph practically alone, the experience was considered by him as the most enriching one in his formation at Columbia. According to Henkin, then began to take form some of the ideas that became the starting-point of his doctoral dissertation.

In 1940, Henkin decided to apply for admission to a doctoral program, without having fully defined what path to follow in his research. He was accepted to three universities, from which he chose Princeton, since the renowned logician Alonzo Church was there, although at the time Henkin was unaware of his work.

Postgraduate Studies
Henkin began his graduate studies at Princeton in 1941, studying under the direction of Church. The Ph.D. program he attended consisted of two years of mathematics courses, after which he was to take a "qualifying" oral examination to show he was well educated in at least three branches of mathematics; with this he would receive a M.A. degree. He would then have another two years to write a doctoral dissertation containing an original research, after which he would get the degree of Ph.D.

The first two years he took courses in logic -taught by Church-, analysis and general topology. In the first logic course with Church were studied several formal systems of Propositional Logic and First-Order Logic; some proofs of completeness and discussed part of the Löwenheim-Skolem theorems were revised, as well as a presentation of Gödel's proof on the completeness of First-Order Logic. In the second one they dealt in great detail with a Second-Order system for Peano Arithmetic, as well as with the incompleteness of this axiomatic theory and the consequent incompleteness of Second-Order Logic.

In 1941 the United States entered the Second World War, altering Henkin's plans. He had to rush his oral qualification exam, with which he obtained the degree of M. A. and left Princeton to take part in the Manhattan Project. This interruption would last four years, during which he contributed his mathematical knowledge working on radar problems and in the design of a plant to separate uranium isotopes. Most of his work required numerical analysis to solve partial differential equations. During this period, all of his work and readings on logic were completely suspended.

Once the war was over, Henkin returned to Princeton in 1946, where he was still required to write a dissertation to complete his Ph.D. studies. Upon his return he joined the logic course that Church had begun a month earlier on Frege's theory of "sense and reference". In this course he discovered Church's theory of types, which he found extremely interesting. The questions he asked about it eventually led him to give his proof of the completeness of the theory of types, which he was able to adapt to also give a new proof of the completeness of First-Order Logic. These results, as well as others that other that emerged from the same ideas, came to take part in Henkin's doctoral dissertation, which was titled "The completeness of formal systems", with which he graduated in June 1947. The dissertation itself was not published, although parts of it were rewritten and published in articles, and. Many years later, Henkin wrote the article "The discovery of my completeness proofs", which contains a detailed review of the contents of his dissertation. The procedures used in it have become frequent methods of proofs in various branches of logic.

After the graduation
Having obtained his Ph.D. degree, Henkin spent two more years at Princeton working on post-doctoral studies. During this time, in 1948, he met Ginette Potvin, during a trip to Montreal with his sister Estelle and Princeton mathematics graduate student Harold Kuhn.  Ginette would become his wife in 1950, a half year after Estelle married Harold.  After completing his second year of postdoctoral studies at Princeton in 1949, Leon returned to California, where he entered the mathematics department at the University of Southern California. There he held the position of assistant professor until 1953.

In 1952 Tarski had managed to obtain a permanent position at Berkeley for Henkin. However, Henkin did not want to accept it, as he was sympathetic to the protests recently raised by the controversial oath of allegiance that had been required of university professors since 1950.  Once the oath requirement disappeared, Henkin accepted Tarski's offer and settled in Berkeley in 1953.

His life in Berkeley
From 1953, most of Henkin's academic activity revolved around Berkeley, where he collaborated with a solid research group in Logic. He remained there for almost all his academic life, except for some periods in which he traveled abroad with scholarships and grants of diverse institutes, like the one-year stay he had in Amsterdam or the one in Israel with the Fulbright Research Grants he was awarded (in 1954 and 1979 respectively).

Henkin was always grateful to Tarski, as it was thanks to him that he was able to settle in Berkeley. After Tarski's death in 1983, he wrote in a personal letter: "I write to tell you that Alfred Tarski, who came to Berkeley in 1942 and founded our great Center for the Study of Logic and Foundations, died Wednesday night, at age 82 [...]. It was he who brought me to Berkeley in 1953, so I owe much to him personally as well as scientifically."

Tarski not only offered Henkin a job opportunity, but also provided him with a very fertile interdisciplinary collaborative environment for the development of Logic. Tarski had founded the Center for the Study of Logic and Foundations in Berkeley, but with Henkin's help he was able to bring together a group of logicians, mathematicians and philosophers who formed the Group in Logic and the Methodology of Science, which is still active today. As part of this project they created an interdisciplinary postgraduate program culminating in a Ph.D. Tarski and Henkin boosted the project by organizing important congresses and conferences on Logic, following Tarski's conception of "logic as a common basis for the whole of human knowledge". The intense activity that took place in Berkeley in the 1950s and 1960s on metalogic was largely due to the activity of Tarski and Henkin, both in teaching and research. Many results of what are today crucial to Model Theory came as a result of the academic activity in Berkeley that took place in those years.

Among the research trips that Henkin did throughout the years are his visits to universities in Hanover, Princeton, Colorado, as well as to several European Universities, such as Oxford (in the United Kingdom), and others in Yugoslavia, Spain, Portugal and France. In 1979, with his second Fulbright Grant, Henkin spent a year in Israel, in Haifa, at the Department of Science Education of the Technion University. On this occasion he also visited two universities in Egypt. In 1982 he first visited  Spain. He gave conferences at several universities, including those in Barcelona, Madrid and Seville.

Henkin had an active role in research and teaching, but his activities at the university went far beyond that. In addition to the dedication he put in his teaching as well as and in guiding the Group in Logic and the Methodology of Science, he held some administrative positions; he was director of the Department of Mathematics from 1966 to 1968, and subsequently from 1983 to 1985. One of the activities to which he devoted most energy was the teaching of mathematics, on which he also did some research.

On some occasions Henkin attended to his children's schools to talk to elementary school children about maths, talking to them about "the negative numbers", or "how to subtract by addition". Around that time (about 1960), Henkin began to alternate his research work in mathematics with research work in teaching mathematics; the latter became increasingly frequent.

In 1991 he was granted the title of Professor Emeritus at the University of Berkeley and retired.

Retirement and death
After he retired, Henkin continued to work on math teaching projects. From 1991, he took part on a summer courses program at Mills College intended to give talented women from across the nation education in mathematics in order to prepare them for college. Finally, Ginette and Henkin moved to Oakland, where Henkin died a few years later, in November 2006.

Always kind to his students and colleagues, whom he frequently invited to his home to enjoy evenings with Ginette, he is remembered as a brilliant researcher, a teacher committed to his discipline and a person who showed solidarity with his community.

One of the phrases that best captures the sentiment expressed in various testimonies of his students is that given by Douglas Hofstadter: "I feel very fortunate to have been his graduate student since I learned from him much more than logic. It is his humanity that conquered my heart. I always wish I am not less kind to my graduate students and no less eager to follow their professional growth after graduation than he was to me".

Legacy

Algebra
Henkin's work on algebra focused on cylindric algebras, a subject he investigated together with Alfred Tarski and Donald Monk. Cylindric Algebra provides structures that are to first-order logic what Boolean algebra is to propositional logic. One of the purposes of Henkin and Tarski in promoting algebraic logic was to attract the interest of mathematicians to logic, convinced as they were that logic could provide unifying principles to mathematics:  "In fact we would go so far as to venture a prediction that through logical research there may emerge important unifying principles which will help to give coherence to a mathematics which sometimes seems in danger of becoming infinitely divisible".

According to Monk, Henkin's research on cylindrical algebra can be divided into the following parts: Algebraic Theory, Algebraic Set Theory, Representation Theorems, Non-representable Algebraic Constructions and Applications to Logic.

Completeness Theorems
In 1949 "The completeness of the first order functional calculus" was published, as well as "Completeness in the theory of types" in 1950. Both presented part of the results exposed in the dissertation "The completeness of formal systems" with which Henkin received his Ph.D. degree at Princeton in 1947. One of Henkin's best known results is that of the completeness of First-Order Logic, published in the above-mentioned 1949 article, which appears as the first theorem of the 1947 dissertation. It states the following:Any set  of sentences of  formally consistent in the deductive system of  is satisfiable by a numerable structure .This theorem is nowadays called the 'completeness theorem', since from it the following easily follows:If  is a set of sentences of  and  is semantic consequence of  , then  is deducible from  .This is the strong version of the completeness theorem, from which the weak version is obtained as a corollary. The latter states the result for the particular case in which  is the empty set, this is to say, the deductive calculus of first order logic is capable of deriving all valid formulas. The weak version, known as Gödel's completeness theorem, had been proved by Gödel in 1929, in his own doctoral thesis. Henkin's proof is more general, more accessible than Gödel's and more easily generalizable to languages of any cardinality. It approaches completeness from a new and fruitful perspective and its greatest quality is perhaps that its proof can be easily adapted to prove the completeness of other deductive systems. Other results central to model theory are obtained as corollaries of the strong completeness of the first order logic proved by Henkin. From it follows, for example, the following result for a first order language  :Every set of well-formed formulas of  that is satisfiable in a −structure is satisfiable in an infinite numerable structure.This result is known as the "downwards" Löwenheim-Skolem theorem. One other result obtained from the completeness theorem is: A set  of well-formed formulas of  has a model if and only if each finite subset of it has a model. The latter is known as the "compactness theorem" of first-order logic, which can also be phrased as: "Any set of well formed formulas of  that is finitely satisfiable is satisfiable". This is to say, if for each of the finite subsets of  there is a structure in which all of its formulas are true, then there is also a structure in which all the formulas of  are true. It is known as "compactness theorem" because it corresponds to the compactness of a certain topological space, defined from semantic notions.

Among the other theorems of completeness given by Henkin, the most relevant is perhaps that of the completeness of Church's Theory of Types, which is the first of the completeness theorems Henkin proved. Then, he adapted the method developed in that proof to prove the completeness of other deductive systems. This method has continued to be used to give proofs of completeness in both classical and non-classical logics, and it has become the usual proof of completeness for First-Order Logic in Logic textbooks. When Henkin published this result in 1949, completeness was not even part of the canonical subjects covered by the textbooks; some twenty years later, this theorem, along with its proof and corollaries, was part of virtually every Logic textbook. As for non-classical logics, Henkin's method can be used, among other things, to extend the completeness of Fuzzy Logic from first order to higher order, producing a complete Fuzzy Type Theory; it also offers a way to obtain results that link classical logic with intuitionist logic; and it allows one to test results of completeness in other non-classical logics, as in the cases of Hybrid Type Theory and Equational Hybrid Propositional Type Theory.

The Discovery of the Completeness Theorems
Despite being one of his best known results, Henkin got to the proof of the completeness of first-order logic "accidentally", trying to prove a completely different result. The order of publication of his articles and even the order of presentation of the theorems in his 1947 dissertation does not reflect the evolution that followed the ideas that led him to his completeness results. However, Henkin simplifies the difficult task of tracing the development and shaping of his ideas by his article "The discovery of my completeness proofs", published in 1996. In it, he describes the process of the development of his dissertation. He doesn't only explain the content of his work, but he also explains the ideas that led to it, from his first logic courses in College until the end of the writing of his thesis.

At the end of the war, Henkin returned to Princeton to complete his doctoral studies, for which he still had to write a dissertation containing an original research. As soon as he arrived at Princeton, he attended Church's course in logic that had begun one month earlier, which dealt with Frege's theory of "sense and reference". Motivated by Frege's ideas, Church wanted to put them into practice through a formal axiomatic theory. To do so, he took the simple Theory of Types he had published a few years earlier, and supplied it with a hierarchy of types, inspired by the idea of "sense" exposed by Frege. It was in this course that Henkin became acquainted with Church's Theory of Types, which he found of great interest. He immediately made a conjecture about it, whose proof he hoped could become his doctoral dissertation.

One of the attributes that drew Henkin's attention to Church's Theory of Types was that the -operator allowed to name many objects in the type hierarchy. As he explains in "The discovery of my completeness proofs", he set out to find out which elements had names in this theory. He began by exploring the elements that were named in the two domains at the base of the type hierarchy. He took  as the universe of individuals, and added a constant for each the number  and the successor function , so that each element in the domain was named from  and repeated occurrences of . Going up through the hierarchy, he tried to specify which functions over those elements were nameable. The set of them was supernumerable, so there had to be some without a name, since there is only a numerable number of expressions. How could be said which elements were the nameable ones? To make each expression correspond to the element it denotated, he needed a choice function, in whose search Henkin invested many efforts. Finally, he realized that by means of the deductive calculus he could form equivalence classes of expressions whose equality could be derived by the calculus, and form with these classes a model isomorphic to the new hierarchy of types formed by the named elements. He had been focusing on the interpretations of the formal language, when the key to solving the problem lied on the deductive system. It remained to make the universe of the objects named by the propositions a set of two elements: the truth values. This could be achieved by expanding the axioms to form a maximally consistent set. Once this was achieved, it could be proved that every consistent set of formulas  has a model that satisfies exactly the formulas of  –the elements of such model are the equivalence classes of the expressions themselves–. That is, he would have managed to give a proof of the completeness of the deductive calculus.

The same method used to prove the completeness of Church's Theory of Types could easily be adapted to give a proof of (strong) completeness of First-Order Logic, and of others that followed later on. The ideas on the nameable elements in the hierarchy of types underlying the discovery of Henkin's completeness proofs led to the successful introduction of new semantics, called general semantics, which are based on general models (or Henkin models).

Henkin's method
Henkin's method to give the completeness proofs consists on building a certain model: it starts with a set of formulas , of which the consistency is assumed. A model is then constructed, which satisfies exactly the formulas of . Henkin's idea to build a suitable model relies on obtaining a sufficiently detailed description of such model using the sentences of the formal language, and to establish which objects could be the elements of such model. If it were known, for each formula of the language of , if it should be satisfied or not by the model, we would have a comprehensive description of the model that would allow its construction. This is exactly what is being looked for: a set of sentences  containing  for which it holds that every sentence of the language or its negation belongs to Gamma. In the case of first order logic one more thing is required: that the set  be exemplified, this is, for all existential formula there is a constant that acts as a witness of it. On the other hand, since the nature of the objects that make up the model's universe is irrelevant, no objection arises against taking as individuals the terms of the language themselves –or classes of equivalence of them–.

The first step that must be taken is to extend the language of  adding an infinite collection of new individual constants, and then to order the formulas of the language (which are infinite). Once this is done, the aim is to inductively construct an infinite chain of consistent and exemplified sets: we start from , systematically adding to this set every formula that doesn't make the resulting set inconsistent, adding also exemplifications of the existential formulas. Thus, an infinite chain of consistent and exemplified sets is built, whose union is a maximally consistent and exemplified set; this will be the required set .

Having achieved to construct this maximally consistent and exemplified set, the model described by it can be constructed. Which individuals constitute the model's universe? In the case of First-Order Logic without equality, the elements of the domain will be the terms of the formal language. To construct the functions and relations of the model we follow thoroughly what  dictates: if the language contains a -relator , its interpretation in the model will be a relationship formed by all the -tuples of terms in the model's universe such that the formula that says they are related belongs to . If the language includes equality, the domain of the model are classes of equivalence of the terms of the language instead. The equivalence relation is established by the formulas of the maximally consistent set: two terms are equal if there is in  a formula stating they are.

Summarizing, the demonstration in the case of a numerable language has two parts:

 Extending the set  to a maximally consistent and exemplified set.
 Constructing the model described by the formulae of this set using the terms of the language –or its equivalence classes– as objects of the model's universe.

General models
The simple Theory of Types, with the -calculus and the standard semantics is sufficiently rich to express arithmetic categorically, from where it follows, by Gödel's incompleteness theorem, that it is incomplete. Following the idea of identifying the namable elements in the hierarchy of types, Henkin proposed a change in the interpretation of the language, accepting as types hierarchies some that  previously were not admitted. If it was asked from each level of the hierarchy not that there must be all the corresponding functions, but only those that are definable, then a new semantics is obtained, and with it a new logic. The resulting semantics is known as general semantics. In it the structures that are admissible as models are those known as 'general models'. These can be used not only in Type Theory, but also, for instance, to obtain complete (and compact) Higher-Order Logics.

Obtaining complete Higher-Order Logics by the use of general semantics meets the expected balance between the expressive power of a logic and the power of its deductive calculus. In Second-Order Logic with standard semantics it is known that quantifying over predicative variables gives the language an immense expressive power, in exchange for which the power of deductive calculus is lost: the latter is not enough to produce the extense set of valid formulas of this logic (with standard semantics). Changing the calculus does not solve anything, since Gödel's incompleteness theorem ensures that no deductive calculus could achieve completeness. On the contrary, by changing the semantics, that is, by changing the sets that form the universes in which the predicative variables and constants are interpreted, the logic turns out to be complete, at the cost of losing expressive capacity.

In Second-Order Logic the set of valid formulas is so large because the concept of standard structure is too restrictive and there are not enough of them to find models that refute the formulas. By relaxing the conditions we ask of the structures on which the language is interpreted, there are more models in which the formulas must be true to be valid and therefore the set of valid formulas is reduced; it does so in such a way that it coincides with the set produced by a deductive calculus, giving rise to completeness.

Towards a translation between logics
One of the areas in which the foundations laid by Henkin's work have proved fruitful is in the search for a logic that works as a common framework for translation between logics. This framework is intended to be used as a metalogical tool; its purpose is not to choose "one logic" above the others, which would suppress the richness provided by the diversity of them, but to provide the adequate context to contrast them, understand them and thus make the best use of the qualities of each one.

A research that takes Henkin's ideas in this direction is that of María Manzano, one of his students, whose proposal is to use Many-Sorted Logic as a common framework for the translation of logics. The aims of this proposal can be synthesized into two: 1) to use a single deductive calculus for all of them; and 2) to use the metaproperties of Many-Sorted Logic in order to more easily proof metaproperties of other logics. In addition, having a logical framework is useful for comparing different logics by comparing the theories that represent them. Although Henkin does not speak of formulae translation, nor does he make explicit a Many-Sorted Language or calculus, the ideas he uses in two of his articles serve as a basis for the approach to translation: "Completeness in the theory of types" and "Banishing the Rule of Substitution for Functional Variables".

Mathematical Induction
The topic of mathematical induction was frequently addressed in Henkin's activities on teaching. Probably his experience in this field was the result of his article "On mathematical induction". This was Henkin's favorite article of his own, of which he even wrote that he considered it his best expository article. In it he defined Peano Models as those that fulfill Peano's three Second-Order Axioms and Induction Models as those that satisfy the third of them: the induction axiom. He demonstrated that although all recursive operations can be introduced in the Peano models, this is not the case in the Induction Models. Concretely, there are Induction Models in which the exponentiation operation cannot be defined. In this article, Henkin also presents the mathematical structure that Induction models can have, which is quite simple: they can either be the standard model, that is, isomorphic to natural numbers, or in two more ways; isomorphic to cycles –which correspond to the  integers module ; or isomorphsicn to what Henkin called "spoons," which is a combination of a finite list followed by a cycle.

Philosophical position
Of the articles published by Henkin, the most philosophical is "Some Notes on Nominalism", which he wrote in response to two articles on nominalism, one by Quine and the other jointly written by Quine and Goodman. The discussions relevant to this philosophical doctrine arise naturally in the proofs of completeness given by Henkin, as well as in his proposal for a change in semantics through general models. Both from the content of his works and from his own statements it is considered that his position was nominalist.

Teaching
Henkin's activity as a university professor was vigorous. He taught at all levels, putting the same care and dedication into each of them. Some of the courses he taught were directly related to his research area, such as "Mathematical Logic", "Metamathematics" or "Cylindric Algebra", but others extended to a great diversity of areas, including, among others, "Fundaments of Geometry", "Algebra and Trigonometry", "Finite Mathematics", "Calculus with Analytic Geometry" or "Mathematical Concepts for Elementary School Teachers". His students agree that his explanations were extremely clear and caught the listener's attention. In the words of one of his students, "part of his magic was his elegant expression of the mathematics, but he also worked hard to engage his audience in conjecturing and seeing the next step or in being surprised by it. He certainly captured the interest of his audiences."

One of the aspects of his lectures in which he put special care was in finding an appropriate pace, facing the constant dilemma of how to find the optimal speed for learning. He considered it important that the students could follow the rhythm of the class, even if this meant that some would found it slow –they could continue at their own pace with the readings. However, he also considered that what was easily learned was easily forgotten, so he sought a balance between making his classes accessible and challenging for students, so that they would make the effort to learn more deeply. About his own experience as a student, he commented in an interview: "That easy way in which ideas came made it too easy to forget them. I probably learned more densely condensed material in what we called the 'seminar for babies in conjunctive topology', conducted by Arthure Stone. I learned more because it forced us to do all the work."

In addition to his courses and supervision of graduate students, Henkin's role in the scholars education was significant. Tarski had invited him to Berkeley with a clear purpose. As a mathematician, Henkin had a key role in Tarski's project to make Berkeley a center of development of logic, bringing together mathematicians, logicians and philosophers. Henkin aided him to carry out the project, helping him in the creation of the interdisciplinary Group in Logic and the Methodology of Science, whose successful performance was largely due to Henkin's drive. Part of this project was the creation of an interdisciplinary university program that culminated in a Ph.D. in "Logic, Methodology and Philosophy of Science". He also collaborated in organising important meetings and conferences that promoted interdisciplinary collaboration united by logic. The outcome was that in the 1950s and 1960s there was a vibrant development of logic in Berkeley, from which many advances in Model Theory emerged.

Although Henkin's first encounter with teaching mathematics was as a professor, later in life he began to do research in mathematics' teaching as well. Some of his writings in this field are: "Retracing Elementary Mathematics", "New directions in secondary school mathematics" or "The roles of action and of thought in mathematics education". From 1979 onwards he put special emphasis on this facet of his research and the last doctoral theses he directed are related to the teaching of mathematics or the integration of minority groups in research.

Henkin liked to write expository articles, for some of which he received awards such as the Chauvenet Prize (1964), for the article "Are Logic and Mathematics Identical? " or the Lester R. Ford Award, for the article "Mathematical Foundations of Mathematics".

Social projects
Throughout his life, Leon Henkin showed a deep commitment to society and was often called a social activist. Many of his mathematics teaching projects sought to bring minority or socially disadvantaged groups closer to mathematics and related areas. He was aware that we are part of history and the context around us, as one of his writings records:"Waves of history wash over our nation, stirring up our society and our institutions. Soon we see changes in the way that all of us do things, including our mathematics and our teaching. These changes form themselves into rivulets and streams that merge at various angles with those arising in parts of our society quite different from education, mathematics, or science. Rivers are formed, contributing powerful currents that will produce future waves of history.

The Great Depression and World War II formed the background of my years of study; the Cold War and the Civil Rights Movement were the backdrop against which I began my career as a research mathematician, and later began to involve myself with mathematics education."Henkin was convinced that changes could be achieved through education and, true to his idea, he committed himself to both elementary mathematics education programs and to programs whose aim was to combat exclusion. He showed a political commitment to society, defending progressive ideas. He inspired many of his students to become involved in mathematics education. Diane Resek, one of his students with an affinity for teaching, described him as follows: "Leon was committed to work toward equity in society. He was able to see that profes- sional mathematicians could make a difference, particularly regarding racial inequities in the United States. He was one of the first people to say that one thing holding back racial minorities and poorer people in America was their low participation rates in math/science careers. He believed that there were ways of teaching and new programs that could correct this problem."Aware of the contributions that mathematicians could make through teaching, Henkin defended that teaching should be valued in the academy environment, as he expressed in a personal letter: "In these times when our traditionally trained mathematics Ph.D.'s are finding rough going in the marketplace, it seems to me that we on the faculty should particularly seek new realms wherein mathematics training can make a substantial contribution to the basic aims of society."

Some of the social projects he formed or participated in are the following. Between 1957 and 1959 he was part of the Summer Institutes, aimed at mathematics teachers and dedicated to improving high school and college education. In 1958 the National Science Foundation authorized the committee of the American Mathematical Society –which had been interested for some years in the use of films and visual material for mathematics education– to produce experimental films for this purpose, accompanied by printed manuals with appendices that would go deeper into the content and problems to be solved. Henkin participated in this project with a film on mathematical induction, whose supplementary manual was printed by the American Mathematical Society. The film was broadcast in the series "Mathematics Today". Between 1961 and 1964, he participated in a series of courses for elementary school teachers, organized by the Committee on the Undergraduate Program in Mathematics. Also around that time, he promoted the Activities to Broaden Opportunity initiative, which sought to provide opportunities for promising students from ethnic minority groups by offering them summer courses and scholarships. He took part in the SEED (Special Elementary Education for the Disadvantaged) program, which encouraged college students to participate in elementary education, as well as in SESAME (Special Excellence in Science and Mathematics Education), the interdisciplinary doctoral program created by members of various science departments, whose purpose was to research teaching and learning of science, engineering, and mathematics. Between 1960 and 1968 he participated in a series of conferences in mathematics schools, and was involved in the development of several films produced by the National Council of Teachers of Mathematics (NCTM). These films dealt with topics such as the integer system and the rational number system. He also participated in support courses for female calculus students and convinced the mathematics department to allow graduate students to receive the same financial support for working as elementary school teachers as they did for working as assistant teachers in college. "He not only believed in equality, but also worked actively to see that it was brought about."

Henkin's main articles

 Henkin, L. (1949). The completeness of the first-order functional calculus. The Journal of Symbolic Logic, 14(3), 159-166.
 Henkin, L. (1950). Completeness in the theory of types. The Journal of Symbolic Logic, 15(2), 81-91.
 Henkin, L. (1953). Banishing the Rule of Substitution for Functional Variables. The Journal of Symbolic Logic, 18(3), 201-208.
 Henkin, L. (1953). Some interconnections between modern algebra and mathematical logic. Transactions of the American Mathematical Society, 74, 410-427.
 Henkin, L. (1953). Some notes on nominalism, The Journal of Symbolic Logic, 18(1), 19-29.
 Henkin, L. (1954) A generalization of the concept of $\omega$-consistency. The Journal of Symbolic Logic. 19(3), 183-196.
 Henkin, L. (1955) The nominalistic interpretation of mathematical language. Bulletin of the Belgian Mathematical Society. 7, 137-141.
 Henkin, L. (1955) The representation theorem for cylindrical algebras. En Skolem, Th., Hasenjaeger, G., Kreisel, G., Robinson, A. (Eds.) Mathematical Interpretation of Formal Systems, pp. 85–97.
 Henkin, L. (1957) A generalization of the concept of -completeness. The Journal of Symbolic Logic. 22(1), 1-14.
 Henkin, L. (1960). On mathematical induction. The American Mathematical Monthly. 67(4), 323-338.
 Henkin, L. (1961). Mathematical Induction. En MAA Film Manual No.1 The Mathematical Association of America, University of Buffalo, Nueva York.
 Henkin, L., Tarski, A. (1961) Cylindric algebras. En Dilworth, R.P. (Ed.) Lattice Theory. Proceedings of Symposia in Pure Mathematics. American Mathematical Society, 2, 83-113.
 Henkin, L. Smith, W. N., Varineau, V. J., Walsh, M. J. (1962) Retracing Elementary Mathematics. Macmillan, New York.
 Henkin, L. (1962). Are logic and mathematics identical?, Science, vol.138, 788-794.
 Henkin, L. (1963). New directions in secondary school mathematics. En Ritchie, R. W. (Ed.) New Directions in Mathematics, 1-6. Prentice Hall, New York.
 Henkin, L. (1963). An Extension of the Craig-Lyndon Interpolation theorem. The Journal of Symbolic Logic. 28(3), 201-216.
 Henkin, L. (1963). A theory of propositional types. Fundamenta mathematicae. 52, 323-344.
 Henkin, L. (1971). Mathematical foundations for mathematics. The American Mathematical Monthly. 78(5), 463-487.
 Henkin, L. (1975). Identity as a logical primitive. Philosophia 5, 31-45.
 Henkin, L. (1977). The logic of equality.The American Mathematical Monthly. 84(8), 597-612.
 Henkin, L. (1995). The roles of action and of thought in mathematics education –one mathematician's passage. Fisher, N.D., Keynes, H.B., Wagreich, Ph.D. (Eds.), Changing the Culture: Mathematics Education in the Research Community, CBMS Issues in Mathematics Education, vol. 5, pp. 3–16. American Mathematical Society in cooperation with Mathematical Association of America, Providence.
 Henkin, L. (1996). The discovery of my completeness proofs, Bulletin of Symbolic Logic, vol. 2(2), 127-158.

Awards received
1964 — The Chauvenet Prize, Mathematical Association of America award to the author of an outstanding expository article on a mathematical topic by a member of the Association.
1972 — Lester R. Ford Award — for Mathematical foundations for mathematics, American Mathematical Monthly 78 (1971), 463–487.
1990 — First recipient of the Gung and Hu Award for Distinguished Service to Mathematics.
1991 — Berkeley Citation — the highest honor/award bestowed by the University of California.
2000 — Leon Henkin Citation — for Distinguished Service, which is presented to a (UC) faculty member for "exceptional commitment to the educational development of students from groups who are underrepresented in the academy."

See also
 Branching quantifier

References

Further reading

Henkin, Leon (1949). "The Completeness of the First-Order Functional Calculus", Journal of Symbolic Logic. 14: 159–166.
Henkin, Leon (1949). "Fragments of the propositional calculus", The Journal of Symbolic Logic 14: 42–48.
Henkin, Leon (1950). "Completeness in the theory of types", Journal of Symbolic Logic 15: 81–91.
Henkin, Leon (1996). «The Discovery of My Completeness Proofs». Bulletin of Symbolic Logic 2 (2): 127-158. ISSN 1079-8986. doi:10.2307/421107.
Manzano, María, Sain, Ildikó, Alonso, Enrique (Eds.). (2014). "The Life and Work of Leon Henkin", Birkhauser.

External links

 Berkeley Citation Award
 An interview with Henkin and others about their experiences at Princeton
 An interview with Henkin about his experience at Princeton
 Leon Henkin, advocate for diversity in math & science, has died by Robert Sanders, UC Berkeley News press release, November 9, 2006.
 Obituaries: Leon Henkin, 85: professor steered minorities and women to math by Valerie J. Nelson, Los Angeles Times, November 16, 2006, p. B-6.
 Leon A. Henkin—Cal math educator by Rick DelVecchio, San Francisco Chronicle, November 20, 2006, p. B-3.
 In Memoriam: Leon Albert Henkin by John Addison, William Craig, Carolyn Kane, and Alan Schoenfeld (University of California Academic Senate memorial).
 In Memoriam: Leon Albert Henkin, 1921–2006 by J. Donald Monk, The Bulletin of Symbolic Logic, vol. 15, no. 3 (Sept. 2009), pp. 326–331.

1921 births
2006 deaths
20th-century American mathematicians
21st-century American mathematicians
Columbia College (New York) alumni
Princeton University alumni
University of California, Berkeley College of Letters and Science faculty
Henkin|Henkin, Leon
American people of Russian-Jewish descent
Manhattan Project people
People from Brooklyn
Mathematicians from New York (state)


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text:
Q: Find only circular links in Neo4j I'm new to Neo4j and have been playing with an idea about people moving houses, in order to learn more about cypher. This is what I have currently
Each Person [:OWNS] a House
Each House [:ISIN] a Street
A Person [:WANTS] (to live in) a Street
The aim is to find a complete 'chain'
If I run
MATCH (s:Street)<-[:WANTS]-(p:Person)-[:OWNS]->(h:House) RETURN s,h,p
This returns me the complete chain linking right back to the person.
What I'm trying to do is only return complete chains and not broken ones.
I've also tried
MATCH (s:Street)<-[:WANTS]-(p:Person)-[:OWNS]->(h:House)-[:ISIN]->(s) RETURN s,h,p
but this never returns results. Any thoughts?
UPDATE
I got the last query returning results by doing this
MATCH (s:Street)<-[:WANTS]-(p:Person)-[:OWNS]->(h:House)-[:ISIN]->(s1: Street) RETURN s,h,p
But am not sure if this is what I want.
I just want to return circular results so I can see complete house moving chains. Ultimately based on one person so I'll need to put a WHERE in there.
I will try move queries tomorrow with a larger dataset

A: MATCH won't repeat a node in a chain, aka a single match can't match the same node in two variables. But you can just break out the last chain to a where clause
MATCH (s:Street)<-[:WANTS]-(p:Person)-[:OWNS]->(h:House)
WHERE (h)-[:ISIN]->(s)
RETURN s,h,p

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