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Results tagged with set-theory
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user 3106
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
15
votes
Examples of mathematical theories that are naturally written in exotic logics
It is up for debate how "natural" their approach is, but Smullyan and Fitting use modal logic to develop forcing, in their book, Set Theory and the Continuum Problem. See Michael Weiss's notes for a d …
- 82.6k
7
votes
Implicit uses of Countable or Dependent Choice
Halmos is not alone in proving the countable additivity of Lebesgue measure without explicitly mentioning (countable) choice. I have the third edition of H. L. Royden's Real Analysis in front of me. I …
16
votes
1
answer
1k
views
Proving that ZF is Artemov-consistent
As discussed in another MO question, Sergei Artemov has proposed that the standard formalization Con(PA) of "PA is consistent" is flawed, and has proposed a different way to formalize "proving that PA …
- 82.6k
5
votes
Why should I believe Martin's Maximum++?
Although it's not really a scholarly publication, the Quanta Magazine article, To Settle Infinity Dispute, a New Law of Logic, gives a good introduction to the topic. That article mentions a conferenc …
- 82.6k
10
votes
In what sense does the sentence $\operatorname{con}(\mathsf{PA})$ "say" that $\mathsf{PA}$ i...
There is really nothing peculiar about Con(PA) in this regard. Let's take a simpler statement, such as
$$(\exists x \exists y \exists z : xxx + yyy - zzz = 114) \vee (\exists x \exists y \exists z : …
- 82.6k
15
votes
Why is an internal proof of consistency satisfactory for some systems?
The answer by user57888 is correct, but let me emphasize two things. The first is that much of the interest in this type of question predates Gödel's theorems. So if you want to understand the origina …
- 82.6k
9
votes
Standard models of N and R: An Alice/Bob approach
Like Burak, I am responding to the OP's request to promote my comments to an answer, with the caveat that I want to avoid wading too deeply into philosophical debates that I think are beyond the scope …
- 82.6k
6
votes
Accepted
What is lost in General Relativity without Hahn-Banach axiom in the ZF+HB set theory?
As Ryan Budney mentioned in a comment, there is some ambiguity about what exactly you mean by "general relativity." General relativity is primarily a physical theory rather than a mathematical theory. …
- 82.6k
6
votes
Unnecessary uses of the axiom of choice
The highly-upvoted, accepted answer (by Theo Johnson-Freyd) to another MO question, Why worry about the axiom of choice?, points out that the usual proof of the Poincaré–Birkhoff–Witt theorem assumes …
10
votes
0
answers
354
views
Feferman's universes for proof assistants?
This question was prompted by a discussion from another MO question about the consistency of ZFC. There are some mathematicians who are comfortable with ZFC but uneasy with large cardinals. For them …
- 82.6k
23
votes
Does anyone still seriously doubt the consistency of $ZFC$?
I interpret the question factually: Do there exist professional mathematicians who seriously doubt the consistency of ZFC? The answer is yes. Here are two examples (though sadly, both mathematicians i …
9
votes
2
answers
657
views
Partitioning a set of cardinality $\kappa$ into more than $\kappa$ disjoint subsets
There is an old result due to Mycielski and Sierpiński, and popularized in a Monthly article by Taylor and Wagon (A Paradox Arising from the Elimination of a Paradox; see also this MO answer), that ca …
- 82.6k
22
votes
What can be preserved in mathematics if all constructions are carried out in ZF?
I agree with Asaf Karagila that the question as literally stated is a bit too sprawling, but you might want to start with Simpson's book, Subsystems of Second-Order Arithmetic. Its goals aren't the s …
- 82.6k
21
votes
History of (proposal of) set-theoretic foundations
I'm not sure why you expect there to be a crisp answer to such a broad question. The SEP article you cited demonstrates that, like most historical questions, the answer is messy and complicated.
Your …
- 82.6k
13
votes
Integration in the surreal numbers
In a recent article in the Notices of the AMS, Philip Ehrlich briefly describes some progress in this area. Below is a relevant excerpt from the article.
Conway originally expressed doubt that “reaso …
- 82.6k