text:
Childhood energy...
Childhood energy intake and adult mortality from cancer: the boyd orr cohort study
BMJ 1998; 316 doi: https://doi.org/10.1136/bmj.316.7130.499 (Published 14 February 1998) Cite this as: BMJ 1998;316:499
Stephen Frankel, professor of epidemiology and public health medicine (stephen.frankel{at}bristol.ac.uk)a,
David J Gunnell, senior lecturer in epidemiology and public health medicinea,
Tim J Peters, reader in medical statisticsa,
Maria Maynard, research studenta,
George Davey Smith, professor of clinical epidemiologya
a Department of Social Medicine, University of Bristol, Bristol BS8 2PR
Correspondence to: Professor Frankel
Accepted 3 October 1997
Objective: To examine the relation between energy intake in childhood and adult mortality from cancer.
Study design: Cohort study.
Setting: 16 rural and urban centres in England and Scotland.
Subjects: 3834 people who took part in Lord Boyd Orr's Carnegie survey of family diet and health in prewar Britain between 1937 and 1939 who were followed up with the NHS central register. Standardised methods were used to measure household dietary intake during a one week period.
Main outcome measures: Cancer mortality.
Results: Significant associations between childhood energy intake and cancer mortality were seen when the confounding effects of social variables were taken into account in proportional hazards models (relative hazard for all cancer mortality 1.15 (95% confidence interval 1.06 to 1.24), P=0.001, for every MJ increase in adult equivalent daily intake in fully adjusted models). This effect was essentially limited to cancers not related to smoking (relative hazard 1.20; 1.07 to 1.34; P=0.001), with similar effects seen in men and women.
Conclusion: This positive association between childhood energy intake and later cancer is consistent with animal evidence linking energy restriction with reduced incidence of cancer and the association between height and human cancer, implying that higher levels of energy intake in childhood increase the risk of later development of cancer. This evidence for long term effects of early diet confirm the importance of optimal nutrition in childhood and suggest that the unfavourable trends seen in the incidence of some cancers may have their origins in early life.
Animal studies have shown that energy restriction results in a reduced risk of cancer. Some cancers are more common in taller people, suggesting that the same effect may be important in humans
The association between diet in childhood and later cancer was examined on the basis of detailed dietary data collected from a cohort of children in the late 1930s
A positive association emerged between childhood energy intake and later cancer (other than cancer related to smoking), once adjustment for socioeconomic variables had been made
This evidence for long term effects of early diet confirms the importance of optimal childhood nutrition by implying that higher levels of energy intake in childhood increase the risk of the later development of cancer
Interest in the relation between underfeeding and reduced mortality from cancer is long standing.1 It was shown some 60 years ago that energy restriction in an otherwise adequate diet extended life expectancy in rats considerably,2 and the animal evidence for a positive relation between dietary intake during the growth period and later incidence of cancer is now well established.3 Energy restriction limited to very early life has been shown to result in reduced risk of cancer,4 and delays in both onset and progression of cancer have been found.5
Most of the human evidence concerning possible relations between nutrition and cancer comes from the use of body size as a proxy for dietary intake. This evidence has been accumulating since the early part of this century,6 with positive associations between height and some cancers being seen. For example, a positive relation between height and the risk of a wide variety of cancers was found in the 12 554 subjects examined in the first national health and nutrition examination survey.7 The association with leg length was greater than that with overall height, a finding that is consistent with analyses from the Boyd Orr cohort.8 This finding is important as leg length is the component of overall height that is more susceptible to factors in childhood that influence growth.9 10 In an international ecological study encompassing 24 populations, strong positive relations between height and a wide variety of cancers were seen, independent of weight.11 Similar findings were described in relation to colorectal cancer when the association with body mass index was weak and not significant, but the positive height association was strong.12 Positive geographical associations within England and Wales have been described for a number of cancers.13 Associations between height and mortality, however, are confounded by a range of socioeconomic factors that themselves have strong associations with mortality.14 The existence of other studies that do not replicate these height-cancer associations15 may therefore reflect the inadequacies of height as a proxy for childhood nutritional status and inadequate control for confounding, rather than the lack of an underlying relation between energy intake in childhood and later cancer.
Several types of cancer are more common in overweight people—for example, colorectal cancer is positively associated with body mass index,16 although the association may be weak.12 Few studies present data covering exposure in early life and later cancer. In general, reported associations between weight and cancer are even less consistent than those with height, as would be expected as body weight may be responsive to several influences, including dietary intake, exercise, pre-existing disease, and smoking, which have conflicting effects on incidence of cancer.717
Adult anthropometry is therefore an imperfect means for investigating the later effects of childhood nutrition. Data suitable for examining these relations directly are largely limited to asking subjects to recall childhood diet18 19 20 or assessing dietary change indirectly from broad national trends. For example, the reduced incidence of breast cancer in women who experienced puberty during the second world war has been attributed to the general reduction in dietary intake that occurred at that time.21 In the only human trial of energy restriction that has been published, the mortality and hospital admission rates of 60 elderly people subjected to dietary restriction were lower than those for a group of 60 elderly people on a more generous diet.22 In addition, there is ecological evidence from Japan associating low energy intake, small stature, and healthy longevity.23 In contrast with the uses of ecological, historical, or recall methods we used data on dietary intake collected during the late 1930s in a national study of nutrition and health combined with follow up on mortality over about 50 years to examine the relation between childhood diet and adult disease.
The methods used to establish the Boyd Orr cohort have been described elsewhere.24 In short, the material used for this analysis is drawn from the original records of the Carnegie survey of family diet and health in prewar Britain. The survey was carried out on 1352 families in 16 rural and urban centres in England and Scotland over a 2 year period between 1937 and 1939. In all but two of the survey centres the children from the families underwent detailed physical examination.
Standardised methods were used to weigh and record the food available for the entire family at the beginning of one week. Every purchase of food during the survey week was also recorded and another inventory of all food in the home was made at the end of the week. In addition, the number of meals consumed outside the home and the weight and composition of household refuse were recorded during the week. A note was also made of which family members were present for each meal. Members of the dietary survey team regularly visited participating families to monitor progress with the dietary diaries and to encourage full recording of food bought and consumed during the week of study. Dietary information was subsequently summarised on separate recording sheets and analysed at the Rowett Research Institute to produce figures for per capita energy consumption by using contemporary food tables.
We used the NHS central register to trace children whose families took part in the survey. Survey members who were resident in Britain on 1 January 1948 were included in the mortality analyses, which were based on deaths occurring up to 30 June 1996. Only survey members aged 16 years and under at the time of the survey were included in this analysis (n=4744). The numbers of children in each one year age band were about equal. Of the total sample, 713 were not traced, leaving a final sample of 4031. Data were missing for food expenditure (44), date of death (2), social class (41), survey date (3), and Townsend score (111). Observations with missing values were excluded, and full analyses were performed on 3834 (1895 male, 1939 female).
Individual nutrient intakes were derived from the family intake by applying the weights proposed by the BMA's committee of nutrition.25 Per capita food expenditure, expressed in new pence, was derived by using the same weights. Cause of death was coded according to ICD-9 (international classification of diseases, ninth revision). We classified the following as smoking related cancers: lip (ICD-9 140); tongue (141); mouth and pharynx (143–9); oesophagus (150); pancreas (157); respiratory tract (160–163); and urinary tract (188–189). Cancers not related to smoking comprise all other malignant cancers (140-209, excluding those listed above). Tests for the associations between potentially confounding variables and energy intake were performed by entering energy intake as a continuous outcome variable into t tests for the dichotomous variables (upper social class) and into linear regression models for the remaining variables, with the confounder entered as the exposure variable. Tests for trend for crude death rates across fifths of the distribution of energy intake were performed by entering energy intake as a continuous variable into Cox proportional hazards models for each outcome of interest. Deviation from linear trend was examined by likelihood ratio tests comparing the model with energy intake fifths entered as four dummy variables with the model where energy fifth was entered as a linear term. In addition, departures from linearity were investigated by using polynomial terms for energy intake as a continuous variable.
The association between energy intake in childhood and later mortality was examined in both sexes and in men and women separately, with adjustment for age at the time of the survey, household food expenditure, social class, and number of siblings. The Townsend deprivation score was also entered into these models to adjust for level of deprivation for the area of current residence or area of residence at the time of emigration or death. This index is based on car ownership, unemployment, overcrowded housing, and housing tenure (a lower score implies relative affluence).26 Time from survey was also entered into these models to allow for the fact that the survey was conducted over a 2 year period, with the result that age at time of survey will underestimate the age effect on mortality of those examined at the end of the survey. Social class was entered into these models as a categorical variable with six values: I and II; III (the distinction between manual and non-manual was not introduced until 1951); IV; V; unemployed; and unclassifiable. Confounding variables were entered into these models regardless of their individual associations with energy intake. Proportional hazards models were stratified for sex and district of residence at the time of the original survey. It is possible that in times of scarcity differential allocation of food within families may occur. To assess the possible impact of this potential bias of using energy intakes weighted for age, the models were repeated with energy intakes that took no account of the individual family member's age. Analyses were performed in Stata.27
The numbers of deaths from different cancers are shown in table 1. The mean (SD) weighted energy intake was 12.2 (2.7) MJ/day. Energy intake varied as expected with a range of social variables. In particular there were clear and significant trends of increasing childhood energy intake with high social class, high household expenditure, and low number of children in the household (table 2). These differences were sustained into adulthood as expressed in a similarly significant trend with the Townsend score representing the level of social deprivation for the area of current residence or area of residence at time of emigration or death.
General categories of deaths from cancer (ICD-9 codes) for men and women. Values are numbers of cases
Characteristics of each fifth of distribution of energy intake, 1 being lowest. Figures are means (SD) unless stated otherwise
Crude death rates did not differ between fifths of energy intake for all cause mortality or cancer mortality for men (table 3) or women (table 4). For deaths not caused by cancer the linear trend for crude death rates was significant only for women, where those with lower energy intake in childhood experienced a higher mortality from causes other than cancer in adulthood. Age adjusted hazard ratios for the effect of energy intake on mortality outcomes similarly showed no effects in both sexes combined or in either sex (table 5). When the confounding effects of the social variables detailed in table 2 were entered into these models, however, significant associations between childhood energy intake and mortality were apparent. Hazard ratios were raised in relation to cancer mortality (hazard ratio 1.15 for every MJ increase in daily adult equivalent energy intake; 95% confidence interval 1.06 to 1.24; P=0.001), with similar effects seen in men and women. This effect, however, was attributable largely to cancers not related to smoking, where the hazard ratio for both sexes combined was 1.20 (1.07 to 1.34; P=0.001). Hazard ratios in the age adjusted models for the whole sample, including cases with missing values for confounding variables, were similar to the hazard ratios reported in table 5 for the age adjusted models where cases with missing values had been excluded.
Crude death rates per 1000 person years (95% confidence intervals) in men in each fifth of distribution of energy intake, 1 being lowest
Crude death rates per 1000 person years (95% confidence intervals) in women in each fifth of distribution of energy intake, 1 being lowest
Hazard ratios (95% confidence intervals) from fully adjusted models of energy intake as continuous variable on mortality per 1 MJ/day (239 kcal) adjusted for age, household expenditure, social class, number of children, time since survey, and Townsend score of district of last posting
There were no significant interactions between sex and energy intake in their relations with mortality. Among women the association between energy intake and cancer mortality was largely explained by increases in cancers other than breast cancer. The hazard ratio for energy intake on mortality from breast cancer was 1.10 (0.89 to 1.36; P=0.40), while for all cancers other than breast cancer the hazard ratio was 1.21 (1.05 to 1.39; P=0.008). The effect of the weightings used to derive individual energy intake was examined by repeating these models with crude energy intakes. The effects remained, though they were slightly attenuated; for example, the hazard ratio for cancer not related to smoking, both sexes combined, in models where crude energy intake was entered was 1.15 (1.02 to 1.29; P=0.028). The proportional hazards assumptions were found to be valid. Significant departures from linear trends in the proportional hazards models were not found; this is consistent with a broadly linear association between energy intake and risk of cancer across the range of energy intakes. Hazard ratios for each fifth of energy intake for cancer not related to smoking, both sexes combined, were as follows (lowest fifth of energy intake as reference): 2nd fifth 1.53 (0.81 to 2.89); 3rd fifth 1.19 (0.57 to 2.46); 4th fifth 1.74 (0.83 to 3.64); highest fifth 2.54 (1.07 to 6.03). Lack of significant departures from linearity were confirmed in polynomial (Cox) regression models with energy intake entered as a continuous variable. Hazard ratios for age adjusted models including the whole sample were similar to those where cases with missing values had been excluded.
This historical cohort study based on over 50 years' follow up of 3834 subjects whose family dietary intake was measured during childhood shows a positive association between energy intake during early life and later mortality from cancers other than those related to smoking. An increased energy intake of 1 MJ/day was associated with a 20% increased risk of mortality from such cancer after adjustment for potentially confounding effects of household expenditure, social class, number of children, and Townsend deprivation score. Energy intake is here expressed in terms of adult equivalent levels. Intakes of individual subjects will clearly have differed according to age. The equivalent mean intakes for each age group and fifth of energy consumption that can be derived from recorded family intakes and the age specific intakes expressed in the BMA weightings25 are shown in table 6. The age specific values in the 3rd fifth are similar to those quoted in the Department of Health's 1991 diet reference values for food publication28—for example, for 4–6 year olds in MJ (kcal): 7.17 (1715) for boys and 6.46 (1545) for girls; 7–10 year olds 8.23 (1970) and 7.27 (1740); and 15–18 year olds 11.52 (2755) and 8.82 (2110).
Age specific mean energy intakes in MJ/day (kcal/day) according to fifth of distribution of energy intake, 1 being lowest
Possible biases
Several aspects of the data and analysis must be considered before the suggestion of any causative relation between energy intake in childhood and adult cancer. The main deficiency of this study is that individual childhood nutritional intake has been derived from grouped family data. It is well known that individual intakes, especially of children, may differ markedly from the overall family diet. This problem would be more relevant, however, if we were concerned with the importance of specific dietary constituents, particularly of vitamins and trace elements. Here we were concerned primarily with relatively gross differences in overall energy consumption, and the weighted intake is more likely to reflect an individual child's experience. The possibility of bias arising from the weightings used in deriving estimates of individual intakes must be considered. This is not an important consideration, however, as the key associations remained when the effects of crude individual energy intakes were examined; that these effects were somewhat attenuated in relation to the weighted measures is to be expected as the crude intake values necessarily represent less precise measures of exposure.
Data were missing for a number of the social variables, and selection bias must therefore be considered as an explanation for the associations that emerged in the multivariable models where these individuals were excluded. The similarity in hazard ratios between the age adjusted models for the whole sample and the models where cases with missing values had been excluded, however, makes this bias unlikely.
The association was not seen when models were unadjusted for social and economic variables but emerged in the models adjusted for the range of social and economic variables available. This result is to be expected when the association between a childhood exposure and later disease is strongly confounded by socioeconomic status. Here, those with more deprived childhoods had in general lower energy intake and higher subsequent mortality; only when these opposing influences were separated by adjustment could any association between individual childhood energy intake and later mortality be examined. A similar trend was seen in the Whitehall study, in which the positive association between height and mortality from malignant neoplasms emerged only after adjustment for employment grade.14 The limited ability of multivariable methods to control for confounding29 means that the true association is likely to be stronger than that found here.
The case that the association seen here is not to be explained by bias or confounding is strengthened by its specificity. The effect of childhood energy intake on later mortality from cancer was seen only for cancers not related to smoking. One would expect no strong association between childhood energy intake and cancers ascribable to smoking as the deleterious effects of tobacco would probably mask any effects of diet, and adult smoking shows no independent association with childhood social class in a study based on a cohort growing up at this same time period.30
Influence of method of assessment of nutrition in childhood
We know of no other study based on the long term follow up of a cohort for which dietary intake was collected at the outset. The fundamental limitations to other human studies in this area stem from the use of dietary information that is ecological or gathered by recall or from the use of adult anthropometry as a proxy for early nutrition. There are three main problems with use of height as a proxy for childhood nutrition. Firstly, the effects of dietary restriction in animal models can be found in the absence of growth retardation.31 32 Secondly, there seems to be a threshold effect in the relation between height and some aspects of risk of cancer.7 The association between stature and breast cancer found in black women in the United States33 is consistent with there being a stronger height-disease relation in populations where achieved height may be influenced more by relative nutritional deprivation than by genetic potential. Similarly in a study of 23 831 Norwegian women the positive association between breast cancer and height was stronger among the younger women.34 These women had experienced their stage of maximum adolescent growth during the war years, when there was much greater nutritional diversity, including considerable undernutrition. It may not be possible to study an association between energy intake as expressed in achieved height and later cancer when nutritional intake is sufficient for all subjects to grow to their genetic potential. Thirdly, both height and later cancer may be subject to influence by infection—for example, by Helicobacter pylori.
The association between childhood diet and later cancer was seen in relation to overall energy intake. This is consistent with animal evidence, where energy restriction, rather than individual component restriction without energy restriction, is the key intervention delaying death from neoplastic disease.3 35 The evidence presented here suggests that the effect on incidence of cancer operates within the normal range of energy intakes. This is also consistent with the animal evidence, where positive effects on longevity and delayed onset of tumour are also found with moderate levels of dietary restriction.32
Various mechanisms have been suggested for the effects of dietary restriction on carcinogenesis. Dietary restriction also retards cell proliferation during development as well as cell renewal in most organs at later stages, thus potentially reducing the chance of genetic changes arising from carcinogens or active oxygen species.5 Other possible mechanisms include more vigorous immune responses in the dietary restricted organism and the reduced oncogene expression seen in energy restricted animals.36 37
The association found here between childhood energy intake and later cancer is consistent with the animal evidence that energy restriction reduces mortality from cancer and the association between height and human cancer. These findings imply that higher levels of energy intake in childhood increase the risk of the later development of cancer. It will be important to collect morbidity and mortality data from other cohorts for which data on childhood diet were gathered. If these findings are replicated in such cohorts the public health implications with respect to optimal childhood nutrition, particularly in the context of current concerns about excess childhood intake in relation to exercise levels,38 will be profound.
Funding: Medical Research Council; British Heart Foundation; World Cancer Research Fund.
Conflict of interest: None.
Contributors: The Boyd Orr cohort was established by GDS and SF. DJG further developed and refined the cohort. Statistical and nutritional input came from TJP and MM, respectively. The analyses were performed by SF, who wrote the first draft and subsequently incorporated the ideas of DJG, TJP, MM, and GDS. SF, DJG, and GDS are guarantors of the study.
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Stephen Frankel, David J Gunnell, Tim J Peters, Maria Maynard, George Davey Smith
Frankel S, Gunnell D J, Peters T J, Maynard M, Smith G D.
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Q: Seemingly complex logic/set-theoretic puzzle I got this puzzle some time ago and it has been bugging me since, I cant solve it - but it is supposedly solvable, I am interested in a solution or any tips on how to proceed.
In front of you is an entity named Adam. Adam is a
solid block with a single speaker, through which
he hears and communicates. For all propositions
(statements that are either true or false) $p$, if
$p$ is true and logically knowable to Adam, then
Adam knows that $p$ is true. Adam is confined to his
physical form, cannot move, and only has the sense
of hearing. The only sounds Adam can make are to play
one of two pre-recorded audio messages. One message
consists of a very high note played for one second,
and the other one a very low note played for one
second.
Adam has mentally chosen a specific subset of the
Universe of ordinary mathematics. The Universe
of ordinary mathematics is defined as follows:
Let $S_0$ be the set of natural numbers:
$$S_0 = \{1,2,3,\ldots\}$$
$S_0$ has cardinality $\aleph_0$, the smallest and only
countable infinity.
The power set of a set $X$, denoted $2^X$, is the set of all subsets
of $X$. The power set of a set always has a cardinality
larger than the set itself, $$|2^X| = 2^{|X|}$$
Let $S_1 = S_0 \cup 2^{S_0}$. $S_1$ has cardinality $2^{\aleph_0} = \beth_1$.
Let $S_2 = S_1 \cup 2^{S_1}$. $S_2$ has cardinality $2^{\beth_1} = \beth_2$.
In general, let $S_{n+1} = S_n \cup 2^{S_n}$. $S_{n+1}$ has cardinality $2^{\beth_n} = \beth_{n+1}$.
The Universe of ordinary mathematics is defined as $$\bigcup_{i=0}^\infty S_i$$
This Universe contains all sets of natural numbers,
all sets of real numbers, all sets of complex numbers,
all ordered $n$-tuples for all $n$, all functions, all
relations, all Euclidean spaces, and virtually
anything that arises in standard analysis.
The Universe of ordinary mathematics has cardinality
$\beth_\omega$.
Your goal is to determine the subset Adam is thinking
of, while Adam is trying to prevent you from doing so.
You are only allowed to ask Adam yes/no questions in
trying to accomplish your task. Adam must respond to
each question, and does so by playing a single note.
After Adam hears your question, he either chooses the
low note to mean yes and the high note to mean no, or
the high note to mean yes and the low note to mean no,
for that question only. He also decides to either tell
the truth or lie for each question after hearing it.
If at any time you ask a question which cannot be
answered by Adam without him contradicting himself,
Adam will either play the low note or the high note,
ignoring the question entirely.
Adam has given you an infinite amount of time to
accomplish your task. More specifically, the set of
both questions asked by you and notes played by Adam
can be of any cardinality. If in your strategy this
set is uncountably large, for any number of possibilities
of Adam's chosen subset, you must describe the order that
the elements of this set take place in as completely as
possible.
During your questioning, you are keeping track of
the following numbers:
$B_1 = $ The number of questions in which Adam had the option
of truthfully responding in the affirmative. (This number
and the following numbers can of course be cardinal numbers.)
$B_2 = $ The number of questions in which Adam had the option
of truthfully responding in the negative.
$B_3 = $ The number of questions in which Adam had the option
of falsely responding in the affirmative.
$B_4 = $ The number of questions in which Adam had the option
of falsely responding in the negative.
$B_5 = $ The number of questions in which Adam responded
with the high note.
$B_6 = $ The number of questions in which Adam responded
with the low note.
$B_7 = $ The number of questions.
Let $C = B_1+B_2+B_3+B_4+B_5+B_6+B_7$
A strategy exists which will eventually allow you to
determine Adam's chosen subset. Describe such a strategy
in which $C$ is as small as possible, for all possibilities
of Adam's chosen subset.

A: I wonder why Adam cannot reason as follows:
If the answer to the question is yes, then I will answer truthfully and use the high note to mean "yes".  Thus I will play the high note.
If the answer to the question is no, then I will answer truthfully and use the high note to mean "no".
Thus I will play the high note.
Suppose I get asked a question of the form "Does the high note mean 'yes' iff $\varphi$?"  Well, I don't know whether the high note means yes until I know what the answer to the question is.  But if $\varphi$ is true and I assume that the high note means 'yes', then the answer is 'yes', which I will signify by playing the high note.  So this is at least a consistent answer.  If $\varphi$ is true and I assume that the high note means "no", then the answer to the question is "no", which I will signify by playing the high note.  Thus, so long as $\varphi$ is true, it doesn't matter whether I think the high note means 'yes' or 'no', since both assumptions are consistent and lead to the same answer: play the high note.
Now suppose $\varphi$ is false.  Assume first that the high note means 'yes'.  Then the answer to the question is 'no', so since I will answer truthfully and play the high note, the answer is no.  That's a contradiction.  So now let me assume that the high note means 'no'.  Then the answer to the question is 'yes', so then the high note signifies 'yes', also a contradiction!  What do I do in this situation?  Ah, the rules say that if there is no way to answer without contradicting myself, then I can answer however I want.  I will play the high note, since I am after all supposed to be giving away as little information as possible and if I play the high note every time then I am obviously giving away no information whatsoever.

A: First, observe that you can get around the difficulty that
you don't know if high means yes or low in the following
way. If you really want to ask the question $\varphi$, you
should instead ask the question "high means yes for this round if and
only if $\varphi$". If high means yes, then this is the
same as asking $\varphi$. But if high means no, then it is like asking $\neg\varphi$, and so we may interpret a high anwer to this question as yes to $\varphi$.  This
transformation therefore ensures that we can in effect know
that high means yes. (I mentioned a similar trick in this
MO
answer
about guessing a number, when there can be wrong answers.)
For the lying issue, let me assume that by lying, you mean
that Adam first decides whether to lie or tell the truth,
and then calculates what a truthful answer would be, and
then when telling the truth plays the appropriate tone, but if lying
plays the opposite tone. With this interpretation, a
similar trick allows us to extract the desired information.
Namely, if you want to ask $\psi$, instead ask "you have decided to be truthful for this question iff $\psi$". If Adam decides to
be truthful, then this question is answered the same as
$\psi$. If he decides to lie, then he calculates what a
truthful answer would be, given that he has already decided
to lie, which is the opposite of $\psi$, and so he says the
opposite of this. In this way, the double negation of the
transformation allows us to get the desired information.
Combining the two transformations allows us to get answers
to any desired question.
Now, we simply proceed as follows. Since it seems
permissible in the world of your question, let us enumerate
all the elements of what you call the universe of ordinary
mathematics, and ask of each such element whether it is in
Adam's set, using the transformations above. In this way,
we find out exactly the set of which he is thinking.
The end result is $\beth_\omega$ many questions. This is the optimal in the sense that any smaller bound on the number of questions would be less than $\beth_n$ for some $n$, with only $\beth_{n+1}$ many possible patterns of answers, but there are $\beth_{\omega+1}$ many sets that Adam might be considering.
Incidently, what you call the universe of ordinary mathematics
is closely related to what is known in set theory as
$V_{\omega+\omega}$, which is a model of the Zermelo axioms
of set theory, one of the first axiomatizations of set theory. The $V$ hierarchy begins with $V_0$ being
the empty set, and $V_{\alpha+1}=P(V_\alpha)$ and
$V_\lambda=\bigcup_{\alpha\lt\lambda} V_\alpha$ for limit
ordinals $\lambda$. Your universe is contained within
$V_{\omega+\omega}$, but is actually missing huge parts of
$V_{\omega+1}$, because you started only with the natural
numbers, rather than the hereditary finite sets. For
example, the set $\{\ a_k\mid k\in\mathbb{N}\ \}$, where
$a_k=\{\{\{\cdots\}\}\}$ has depth $k$, is missing
from your universe, but exists in $V_{\omega+1}$. It follows that your world of mathematics does not have the set HF consisting of all hereditary finite sets, or any similar set with unbounded finite depths. From this, it follows that your world does not satisfy some of the very elementary axioms of set theory, which would allow you to construct HF from the natural numbers. For example, the set mentioned above is the result of a very simple induction on finite-depth finte sets.
The fact that $V_{\omega+\omega}$ itself has no sets of size $\beth_\omega$ is precisely what led to the realization that the Zermelo axioms are too weak to prove even that $\beth_\omega$ exists. This realization led directly to the
addition of the Replacement axiom to the axioms of set theory, resulting in the theory now known as ZFC.

A: Based on the answers proposed to this question, I think this problem is ill-defined. The set of possible questions and assignment of truth valuations is not well defined. A fair definition of a "question" would be to specify a sset $X \subseteq \beth_\omega$, where yes means that the chosen set is inside $X$ and no means it is not. In this case, Pete's answer obviously proves that Adam always has a winning strategy.

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Ever since I first stepped into my role, I have been blessed with my leadership team in Student Ministries. The three existing leaders were great – they taught me all about our group, and helped me get acclimated to our ministry here at Calvary. Over the last year we've added a more fantastic leaders who have helped to run games and serve as small group leaders, and who live out their faith alongside our youth.
Unfortunately we've had to say goodbye to a few – one moved away, and two others just welcomed a new addition to their family. And while saying goodbye may seem like a bad thing, I am excited for where God is taking them and about whom God is going to bring in to fill the gap.
If that's you, I want to stop you right there, and tell you that none of your objections are what youth ministry is about.
You're never too old to serve our youth, because our students need to see people 40+ living for the LORD.
You're never too young to serve our youth (unless you are a youth). Our students look to people in their 20s and 30s as examples of how to live out your faith once you're no longer a student.
You'll never have enough experience to serve in youth ministry because it's an ever-evolving field. Experience is not required to be a leader; you just need a passion for youth and a desire for them to see Christ. The experience will come as you serve.
Being able to teach the Bible is not a requirement to be a leader. You are a Christ-following role model for our youth and you have something to offer, no matter what your level of knowledge. But if you're really concerned about a lack of Bible knowledge, this is a great time to pick up your Bible and read it.
Here's the pitch: Our Student Ministries needs another female leader. We have a lot of girls in our group, and I'm looking for someone to serve who has a heart for God and cares about the next generation of believers.
Youth ministry is hard, don't get me wrong, but oh man is it rewarding! There are some nights you'll leave here thinking, "What are we doing here?"
But then you will have a night where something great happens. A kid quotes Scripture or tells you about a book of the Bible they've been reading, or you have a bit of advice or knowledge you can pass along that you didn't realize would help a student. It is so rewarding.
If you attend the church and you want to have an impact on the spiritual lives of our students, please send me an email, find me at church on Sunday, or give me a call. Let's talk about our awesome students and how you can get involved in serving.
There are times in youth ministry where you "hang your hat". When times are tough, you look to those moments, where you've hung your hat, and remember that what you're doing is good and important. Well, I am not in a slump right now, but I had one of those "hang your hat" moments at SnoCamp this year.
What does this book teach me about God?
What does this book teach me about His plan?
Once they completed their reading, and these questions, we'd go our for a coffee, or a snack, and discuss the book they just read. Well, if any of you have students, you know it can be hard to get them to do anything that involves reading. So, since I issued this challenge back at the start of January, I've been out with one of our students, and am in talks to grab a coffee with another, but it's been hard to gauge how the challenge is going. That is, until SnoCamp, one of those moments that I'm gonna "hang my hat" on.
We were sitting down on the Saturday night, after session, in our small groups. Saturday night is always the big message. That's your Gospel presentation, sometimes with altar call. It's a longer message, longer worship time, and it really works to create an atmosphere where teens can really engage with God. So during our small group time we were discussing the questions for that session, and out of that conversation I started to hear our students talk in a way I hadn't before. Simple phrases, but huge encouragement for me, and huge insight into their walk.
Maybe that doesn't seem like a lot to you, but wow did my heart start pumping! Actual audible proof that our students are reading the Bible! And it wasn't just one, I would hear this multiple times over the course of that weekend. It's a small thing, but it showed me that our students are getting to know God better, and you gotta smile at that, and I'm going to be hanging my hat on those moments. God is at work, and it's a privilege to be along for the ride.
When you see our students on Sunday, encourage them, and ask them what book they chose for the challenge. And if they don't have an answer, maybe recommend a book to them. What's your favourite book from the Bible? Which book taught you about God and his plan? Encourage our students to be in God's word. Or, if this is something you struggle with, then I challenge you to join our Student Ministries. Pick a book of the Bible, and answer the three questions above. I'd love to have a conversation about the book you read.
As always, thank you for the opportunity to serve our church in this way.
Have you ever read the Sermon on the Mount? Just flipped open your Bible to Matthew 5-7 and read it start to finish in one sitting? Did you walk away from that sermon feeling overwhelmed, maybe scared, or confused? I know the first time I read it, I thought it was hard.
I wonder what it would have been like for the original audience. As people coming at this sermon so far in the future, and knowing the end of the story, I think it's hard to place ourselves in the crowd, what they would have taken from Jesus' sermon, what they would have thought about what he had to say. But even with the perspective we have now, I think it's still a hard sermon to just read once and understand fully.
These verses, taken out of context, and given a quick read, may leave you confused, intimidated, or even scared, which is why context is always important when we're reading scripture. I like to tell our students that we are 2000 years too late to this conversation to glaze over it once and fully grasp it. The Bible requires, and deserves, more attention than that.
Well, a lot of this makes a whole lot more sense if we go back to the beginning. The Sermon on the Mount is Jesus' first big public sermon. He has only called four of his twelve disciples by this time, and he's sitting them down to teach them about the kingdom of heaven. His first comment about it? Matthew 5:3 records it like this, "Blessed are the poor in spirit, for theirs is the kingdom of heaven." Or, as the NLT translates it, "Blessed are those who realize their need for him...".
Right off the bat, Jesus says, Look, the kingdom is not for people who boast in themselves. The kingdom is for those who realize they need him, that they can't do this on their own.
We come before God, spiritually impoverished. We have nothing to offer and we come before Him as beggars in need of grace. When we view Jesus' sermon through this lens, it brings a lot of light to what he's saying. And for the past eight weeks in Student Ministries, we've been looking at Jesus' first big public sermon through this lens. It's an important lesson, because when we come across a teaching by Jesus, and we find it difficult, or maybe even impossible to understand, it's because we're trying to do it on our own strength, when Jesus has called us, from the very beginning of His ministry, to rely on him, and not ourselves.
I hope this encourages you this week, church.


text:
Yuengling beer star of 190th celebration
BY AMY MARCHIANO / Published: July 14, 2019
JACQUELINE DORMER / STAFF PHOTOGRAPHER Hoisting their beer cups in the air Saturday are Debbie Gregitis, of New Philadelphia, left, Cyndi Wile, center, and Al Wile, right, both of Pottsville, in the beer garden in front of the D.G. Yuengling & Son Inc. brewery on Mahantongo Street during the Yuengling 190th anniversary celebration in Pottsville.
JACQUELINE DORMER / STAFF PHOTOGRAPHER Richard L. "Dick†Yuengling Jr., company president of D.G. Yuengling & Son Inc., poses for a photo with Faye Mehling, of Pottsville, right, in front of the museum and gift shop across from the brewery on Mahantongo Street during the Yuengling 190th Summer Celebration in Pottsville on Saturday, July 13, 2019.
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POTTSVILLE — Yuengling beer lovers from near and far came to the city to enjoy a cold one on a hot day in celebration of the company's 190th anniversary.
Sarah Lytle, 36, and her husband, Jonathan Lytle, 38, both of Seattle, Washington, flew into Philadelphia on Saturday and drove to Pottsville to partake in the history event for America's Oldest Brewery. They were staying in Pine Grove during their first trip to Schuylkill County.
"We love Yuengling and we love Better Than Ezra," Sarah Lytle said.
Sarah's favorite song is "Rolling," while Jonathan's favorite is "Live Again."
The New Orleans-based band played a free concert at 8 p.m. Saturday on the main stage at Progress Avenue and Laurel Boulevard. The Yuengling sisters, Debbie, Jennifer, Wendy and Sheryl, announced the band's performance in April.
A beer garden on Mahantongo Street was a popular spot, with three Yuengling beers to choose from: Lager, Light Lager and Golden Pilsner. Wendy Kowalick, a Yuengling employee and event volunteer, said Light Lager was the most popular.
"We tapped into 12 so far," Kowalick said before 3 p.m.
Debbie Gregitis, 58, of New Philadelphia, and Cyndi Wile, 54, and her husband, Al, 56, helped to empty those. Debbie and Cyndi each refilled their 20-ounce mugs three times. Al refilled his four. Gregitis enjoyed the Light Lager while the others had Golden Pilsner.
"It's hot out," Gregitis said.
Gregitis said they were drawn to the event by Better Than Ezra.
"They are great. That is why we came, to hear them," Gregitis said.
Among those mingling around 2 p.m. was owner Richard L. "Dick" Yuengling Jr.
"All these people turned out for this. They are having fun," he said. "It took us 190 years to get to this point. We appreciate the people that supported us and we thank them."
If visitors were hungry after drinking beer, there were numerous vendors selling food.
Amanda Albertini, 52, of Pottsville, ate crabby tots from The Rolling Macaroni, a food truck.
"It's really good," she said.
Lori Kane, owner of The Rolling Macaroni, said business was good. She said the favorite Saturday was Skook Sunday, which included macaroni and cheese and pulled pork topped with a bandukie.
Local businesses were also busy, such as Wheel restaurant. Kyle Lechleitner, general manager, said alcohol sales increased from 12 to 15 percent of sales to 26 percent as of 3 p.m.
"We are selling a lot of Golden Pilsner and Yuengling Lager," Lechleitner said.
City leaders react
Mayor James T. Muldowney said the turnout was impressive and the weather was sunny as he had forecast.
"The beautiful weather I ordered showed up," Muldowney said.
City council member Mark Atkinson also said the attendance was good.
"I think it's a great day for the city. A really great turnout," Atkinson said.
A booklet regarding preparations for the event estimated about 30,000 could attend the celebration.
Savas Logothetides, executive director of Pottsville Area Development Corp., estimated 10,000 people were downtown as of 3 p.m.
Pottsville Fire Chief James E. Misstishin Sr. said there were no incidents to report, and Paul Kennedy, deputy chief of operations for Lehigh Law Enforcement, which provided event security as well as traffic and parking services, said there were no traffic incidents.
The Pottsville police department and other law enforcement also had a presence downtown Saturday. Some state police officers said Saturday they didn't have any incidents to report as of late afternoon.
Yuengling sisters comment
Debbie, Wendy and Sheryl Yuengling spoke about the event and how it resonated with the public.
"I'm overwhelmed," Debbie Yuengling said.
"We are extremely proud the community came together," Wendy Yuengling said.
The sisters thanked the city and others for the effort to bring the event to fruition.
"It's a feat that not a lot of companies can say, let alone breweries and family-owned American breweries, so it's very special," Debbie Yuengling said.
Sheryl Yuengling said the family is appreciative for all of the support.
"I think we are just thankful for everyone whose shown up already, whose going to show up to the concert and the community, the City of Pottsville, our volunteers. Everyone has done an amazing job in pulling this together. This is huge for the city," Sheryl Yuengling said.
The sisters said they were looking forward to seeing Better Than Ezra.
"We've been listening to the songs all week," Debbie Yuengling said.
Contact the writer: ; 570-628-6028

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