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Results tagged with set-theory
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user 2926
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.
3
votes
Accepted
Currying a " partial mapping A*B -->_p C
For plain vanilla sets and functions, a partial function $f$ from $A$ to $B$ can be viewed as essentially the same thing as a total function $g: A \to B + 1$, where the codomain is the disjoint union …
- 53.3k
3
votes
How to decompose an infinite set into two isomorphic ones without choice?
The question seems interesting to me. There's an old result of Tarski that the axiom of choice is a consequence of ZF + the assertion "every infinite set $u$ is in bijection with $u^2$". A proof of th …
- 53.3k
9
votes
How much larger is the powerset of a transfinite set?
As you probably know, $\aleph_1$ is (by definition) the first uncountable cardinality, and Cantor's continuum hypothesis is that $2^{\aleph_0} = \aleph_1$. For any ordinal $\alpha$, $\aleph_\alpha$ is …
- 53.3k
28
votes
How would set theory research be affected by using ETCS instead of ZFC?
I like the fact that you asked this question, but I'm a little worried that it will be seen as "subjective and argumentative".
For the second question, I think a lot of set theorists (trained since ch …
- 53.3k
5
votes
Simple adjective for "of the size of a proper class"?
The word that immediately comes to mind is "large". "Large category", etc.
Edit: Carl Mummert suggested this one, which I should have remembered myself and which is definitely widely used: "unbounde …
- 53.3k
3
votes
A question about J.H. Conway's SURREAL NUMBERS
Garabed, I believe the class of surreals can be encoded by a class formula in ZFC.
Surreal numbers are particular kinds of Conway games, and each Conway game can be expressed as a well-founded roote …
- 53.3k
8
votes
Infinitely Noetherian Rings
My own thoughts gravitate to valuation rings. These can be defined in a variety of ways; for example, as an integral domain $O$ such that for any nonzero element $x$ in its field of fractions $K$, eit …
- 53.3k
17
votes
Is there a constructive proof of Cantor–Bernstein–Schroeder theorem ?
If you accept that toposes are models of constructive set theory, then another way to answer the question is to give a (non-Boolean) topos where the CBS theorem fails; that would show that this theore …
- 53.3k
6
votes
Accepted
Curious decomposition between two sets
Yes, I think the exact same proof goes through for binary relations $R: X \nrightarrow Y$ and $S: Y \nrightarrow X$ (which of course includes the partial function case). Each induces a monotone operat …
- 53.3k
9
votes
Well-ordering with a topological property
The answer is no. Assume there is such a well-order; then some downward-closed subset $D$ (which might be an initial segment, or might be all of $\mathbb{R}$) will be of order type $2^{\aleph_0}$, i.e …
- 53.3k
13
votes
How constructive is Doyle-Conway's 'Division by three'?
(I'm cutting and pasting and slightly modifying some comments taken from a discussion on this question currently taking place at the nForum. It's based on my memory of their paper, which I do not have …
- 53.3k
4
votes
Independence and Category Theory
First, may I ask that we please not get into name-calling or ad hominem attacks. You should think of this as a site at a professional level, and the behavior should be more or less that expected at a …
- 53.3k
14
votes
Nonessential use of large cardinals
I think this example given at Richard Borcherds's blog would qualify, no?
- 53.3k
7
votes
Axiom of Choice in a weaker system
A small quibble: I would view these principles as primarily set-theoretic principles, not logical principles per se. It's actually a little bit tricky, because even if we choose to weaken the metalogi …
- 53.3k
10
votes
Accepted
Universal Objects in Big Categories
In ZFC, a class is given by a class formula $\phi$, that is, a formula expressed in the language of ZFC. For example, the notion of group can be formalized by such a formula $\phi$. The notion of grou …
- 53.3k