9
votes
2answers
331 views

How necessary is Godel's Condensation Lemma

It seems that the Godel's Condensation Lemma is typically used to show that certain constructible sets will appear by some stage of the construction of $L$. For example in the proof that CH holds in ...
1
vote
0answers
219 views

If there is a Reinhardt cardinal, then there is one universe? [closed]

If there is a nontrivial elementary embedding $j:V \to V$, then there is a universe which contains all the large cardinals. Is there such a universe? Does this imply there is one universe from ...
2
votes
2answers
1k views

In What Sense is Set Theory a 'Foundation' for Mathematics? [closed]

In what sense is set theory a foundation for mathematics? To my mind (for what that is worth), there are at least three (somewhat) distinct senses in which set 'theory' (I put "theory" in scare ...
0
votes
1answer
106 views

$\epsilon$-Formalization of Undecidability of CH

Can the statement CH is not provable in ZFC be formalized as en $\epsilon$-Formula $\phi$ s.t. $ZFC \vdash \phi $ If so why is it refered to as an "metatheorem".
4
votes
1answer
416 views

Does ZF prove that a finite subtheory axiomatizes it over transitive proper class models?

If $\text{ZF}$ is consistent, then it is not finitely axiomatizable. For if $\Gamma$ is a finite axiomatization, then $\text{ZF}$ proves by reflection that $\Gamma$ has a set model, and hence (since ...
11
votes
3answers
1k views

Are there natural examples of mathematical statements which follow from consistency statements?

Motivation One of the methods for strictly extending a theory $T$ (which is axiomatizable and consistent, and includes enough arithmetic) is adding the sentence expressing the consistency of $T$ ( ...
4
votes
1answer
762 views

Bourbaki theory of isomorphism, examples of untransportable formulas

In their book "Theory of sets" Bourbaki suggested a general theory of isomorphism. (See also http://www.tau.ac.il/~corry/publications/articles/pdf/bourbaki-structures.pdf ) The example of an ...
22
votes
8answers
2k views

Intuitive and/or philosophical explanation for set theory paradoxes

Every student of set theory knows that the early axiomatization of the theory had to deal with spectacular paradoxes such as Russel's, Burali-Forti's etc. This is why the (self-contradictory) ...
6
votes
7answers
1k views

The isomorphism inference rule

Suppose we are writing very detailed proofs, absolutely without any gaps (for example, for checking proofs by computer). In such formal proofs every step (even a trivial one) must be justified. For ...