How much of mathematics did Russell's paradox really break?
According to my quite likely wildly oversimplified understanding, a revolution occurred in the foundations of mathematics when Cantor's formulation of set theory was found to be inconsistent due to Russell's paradox, which ultimately lead to the development of Zermelo-Fraenkel set theory, thus putting mathematics back on solid formal foundations.
My question is, how much of a revolution did this process cause in the rest of mathematics, outside of the formal foundations?
I can imagine two extremes, with the truth presumably lying somewhere in between. On the one hand one could imagine that a crack in the foundations would break the whole of mathematics, with most theorems even in quite applied topics needing to be re-derived along quite different lines within the new system, in a process that would take many years. On the other hand I can imagine it not really making much difference at all, with most of the higher-level results being somehow independent of the low-level stuff below them, so that the old foundations could be swapped out and new ones put in without disturbing the structures that were built on top of them.
If I had to guess, I would say it was closer to the latter, since when reasoning about higher or applied mathematics we rarely have to go right down to the axioms of set theory. But I would appreciate knowing historically how it played out.
NathanielNathaniel
$\begingroup$ Note that Russell's paradox doesn't actually break all that much of even naive set theory; it only comes up when you try to create really big sets. You can do most of combinatorics without straying beyond finite sets, and you can do most of analysis without straying beyond subsets of Euclidean space (or maybe sets of continuous functions on Euclidean space, which also aren't a problem). These sorts of foundational issues really only showed up on working mathematicians' radar screens when algebraists and topologists started to embrace the language of categories and universal constructions. $\endgroup$ – Paul Siegel Sep 20 '15 at 10:19
$\begingroup$ When I heard this paradox for the first time, I was curious: What sort of mathematical entity was Russell thinking in order to obtain this? According to Peter J Cameron, he was motivated by some readings on the theologian Thierry of Chartres. Jesus is always a screw up, see? $\endgroup$ – Billy Rubina Jan 23 '16 at 16:24
$\begingroup$ @BillyRubina That footnote says that he "might have been pre-empted" by Thierry of Chartres. This does not imply that Russell had any awareness of his work. $\endgroup$ – Robert Furber May 31 '18 at 20:54
Your guess is right. Russell's paradox broke only what people considered at that time as foundations of set theory. More specifically, the logical foundation system built by Frege. Of course this was very disturbing, because many people understood that logic and set theory is really the foundation of the whole mathematics. However it did not "invalidate" any theorems outside set theory and some closely connected areas, like the new theory of functions of the real variable. I am sure that most mathematicians, doing for example differential equations or functions of complex variable or group theory or geometry, did not care much about Russell's paradox. Soon, many theories of foundations were developed to avoid Russell's paradox and similar paradoxes. One is due to Russell himself (it is called the theory of types), another is Intuitionism. Eventually most mathematicians settled with ZF system. Intuitionism (which later evolved to Constructive Mathematics) founded by Brouwer was the most radical attempt to save the foundations. It indeed rejected much of the classical mathematics. Discussions about Intuitionism continued well into the second half of the 20s century. But most mathematicians working in other areas than foundations were not really very interested in these discussions.
$\begingroup$ For example it is not true in constructive mathematics, that a bounded increasing sequence of real numbers has a limit. $\endgroup$ – Alexandre Eremenko Sep 18 '15 at 1:44
$\begingroup$ Take any calculus book, and see how much n it depends on this statement. $\endgroup$ – Alexandre Eremenko Sep 18 '15 at 1:45
$\begingroup$ @AlexandreEremenko But if one takes completeness as a defining axiom of the reals - "Every nonempty subset bounded above has a supremum" - doesn't that follow, well, axiomatically? $\endgroup$ – Reinstate Monica Sep 18 '15 at 19:38
$\begingroup$ For example it is not true in constructive mathematics, that a bounded increasing sequence of real numbers has a limit. But I suspect an intuitionist would be able to make a statement that would be the same for all practical purposes. In practical applications, we don't care about the distinction. For example, nobody has ever measured an irrational number in a physics experiment. $\endgroup$ – Ben Crowell Sep 18 '15 at 19:41
$\begingroup$ Mathematics is MUCH larger than its practical applications:-) $\endgroup$ – Alexandre Eremenko Sep 19 '15 at 2:41
[...] due to Russell's paradox [1901], which ultimately led to the development of Zermelo-Fraenkel set theory, thus putting mathematics back on solid formal foundations.
but be aware that Frege's Begriffsschrift is only form 1879! The idea of using $\forall, \exists$ is younger than the photoelectric effect and so is the line of thinking that you should actually use formal logic for mathematics in this way.
Russel's issue arose a few years after the "logic people" took a look into set theory for the first time. What would be the "solid foundations you put mathematics back on" that mathematicians in different fields would already know (much less trust and care) about?
Nikolaj-KNikolaj-K
$\begingroup$ I'm having a hard time parsing your final sentence. But I take your general point as being that mathematics wasn't really seen as being based on formal foundations much before 1901 anyway - that point is well taken, thank you. $\endgroup$ – Nathaniel Oct 5 '15 at 8:47
Nothing of mathematics was disturbed or spoilt by Russell's paradox. Not even a simple lemma. The reason ist that set therory is not foundational to mathematics (1,2) because it even contradicts mathematics (3).
(1) "set theory is largely irrelevant to the practice of most mathematics. Most professional mathematicians never have occasion to use the Zermelo-Fraenkel axioms, while others do not even know them. [Saunders Mac Lane: "Mathematical models: A sketch for the philosophy of mathematics", American Mathematical Monthly, Vol. 88,7 (1981) p. 467f]
(2) "I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come to publish a critique." [T. Skolem: "Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre", Akademiska Bokhandeln, Helsinki (1923) 217-232, reprinted as "Some remarks on axiomatized set theory" in J. van Heijenoort: "From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931", Harvard University Press, Cambridge, Mass. (1967) 290-301]
(3) Here is the gist of an argument made by Mückenheim: Scrooge McDuck every day receives 10 \$ and issues 1 \$. If he issues always the dollars received first and if he applies modern set theory, then he will go bankrupt because each of the dollars received will be spent. The set-limit is empty. According to mathematics we have to take the limit of the cardinality of the dollars held. This limit is infinite and stands in clear contradiction to set theory.
EDIT: Downvotes will not change the facts:
The actual infinite is not required for the mathematics of the physical world. [S. Feferman: "Infinity in mathematics: Is Cantor necessary?" in "In the light of logic", Oxford Univ. Press (1998) p. 30]
In his concluding chapters, Feferman uses tools from the special part of logic called proof theory to explain how the vast part if not all of scientifically applicable mathematics can be justified on the basis of purely arithmetical principles. At least to that extent, the question raised in two of the essays of the volume, "Is Cantor Necessary?", is answered with a resounding "no". [S. Feferman, "In the light of logic", Oxford Univ. Press (1998) description from the jacket flap]
OttoOtto
$\begingroup$ Your point (3) is pretty strange. Just what is a "set-limit"? If you mean the intersection of all the sets, that is indeed empty, but what does that have to do with "he will go bankrupt"? The intersection of infinitely many sets does have a meaning but no one says that meaning is related to Scrooge McDuck's net worth. Avoiding this kind of sloppiness, ambiguity of the result of an infinite process, is something that set theory successfully avoids. $\endgroup$ – Rory Daulton Jun 17 '17 at 10:13
$\begingroup$ @Roy Daulton: The set limit is explained in the essay The little demon: hs-augsburg.de/~mueckenh/Transfinity/The%20little%20demon.pdf. The formulas are given there. It is a simplified version of Fraenkel's explanation of the exhaustion of infinite sets by the story of Tristram Shandy. If McDuck does not go bancrupt then also not all fractions get enumerated by Cantor's enumeration. The point is that the set of dollars not spent as well as the set of fractions not enumerated must be empty. If all dollars are spent then McDuck has no dollars. That means he is bancrupt. $\endgroup$ – Otto Jun 17 '17 at 10:51
$\begingroup$ As far as I can see, your linked article's complaint is that, for the sets $M_k$ and the definition of set-limit, it wants $\lim(\mathrm{Card}(M_k)) = \mathrm{Card}(\lim(M_k))$. But why should that be? Many properties are not preserved in the limit--why should cardinality be preserved? Do you insist that the limit of continuous functions be continuous? McDuck's wealth in eternity is the left hand side of that "equation", the cardinality of the set limit is on the right. Why should they be equal? $\endgroup$ – Rory Daulton Jun 17 '17 at 11:30
$\begingroup$ @Rory Daulton: Maybe that set theorists believe that infinite sets can be exhausted and that the limit of a continuously increasing positive function can be zero. In any case mathematics, namely analysis yields the contrary, namely the left-hand side. And this clearly contradicts the "set limit" at the right-hand side, because McDuck cannot leave all dollars and be rich. Therefore set theory cannot be used as a foundation to derive mathematics from. $\endgroup$ – Otto Jun 17 '17 at 11:44
$\begingroup$ @Rory Daulton: It is not enough to define (by some arbitrarily devised formula) that McDuck gets bankrupt or that the limit of not enumerated fractions is the empty set. We have to prove that all fractions somehow get enumerated. That is impossible using analysis, because analysis proves the contrary. $\endgroup$ – Otto Jun 18 '17 at 9:50
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