Haim
  • Member for 12 years
  • Jerusalem, Israel
Solutions to the Continuum Hypothesis
22 votes

Regarding Shelah's approach, I believe that the following paper should be quite accessible to non-professionals: YOU CAN ENTER CANTOR’S PARADISE! Now, I have no idea how to explain Woodin's ...

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Large cardinals
Accepted answer
10 votes

By the well-known Levy-Solovay theorem, large cardinal properties are preserved under "small" forcing. Therefore CH is an assumption above ZFC which is not settled by large cardinal axioms.

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Propositional Logic, First-Order Logic, and Higher-Order Logics
10 votes

" What are the advantages/disadvantages of using lower/higher-order formal logics?" It seems that "strong" logics are not as well-behaved as first order logic. By Lindstrom's theorem, FOL is the ...

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Complete extensions of first order logic (or language)
7 votes

Shelah has some nice results on complete extensions of first order logic. For example, he introduced cofinality quantifiers in which the truth value of a given formula (Qxy)Phi(x,y,a) is determined by ...

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Forcing as a tool to prove theorems
7 votes

There are also some model-theoretic examples of such proofs (in ZFC). Once you prove that for a given logic L, the notions of consistency and completeness (for theories in L) are absolute, it's ...

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Is there a ground between Set Theory and Group Theory/Algebra?
6 votes

It turns out that when it comes to infinite groups/modules, some algebraic concepts are deeply connected to the underlying set theory (for example, the notion of freeness, the structure of Ext, etc). ...

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What are some mathematical concepts that were (pretty much) created from scratch and do not owe a debt to previous work?
4 votes

Pcf theory/cardinal arithmetic. Well, it's not exactly built from scratch, but there are plenty of nice results which do not use any sophisticated metamathematical machinery (such as forcing, inner ...

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What are the worst notations, in your opinion?
4 votes

p < q as in "the forcing condition p is stronger than q".

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The concept of duality
3 votes

The duality between measure and category in the set theory of the reals.

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Characterizing forcings that don't add any dominating reals
3 votes

There is a somewhat related result by Shelah: Any Suslin ccc forcing which adds a non-dominated real adds a Cohen real. The proof can be found here: http://shelah.logic.at/files/480.pdf

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(Finite) Classification Theory
3 votes

In addition to Andreas' remark on the number of non-isomorphic models, perhaps it's noteworthy to say that many non-structure theorems are aiming at the construction of many models which not only are ...

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How would one even begin to try to prove that a simple number-theoretic statement is undecidable?
2 votes

I would like to extend the above question: It seems that most previous comments are dealing with independence of number theoretic statements from PA. I would like to know your input regarding the ...

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Cardinality of the set of all paths in the infinite complete infinitary tree
1 votes

You may be also interested in the following paper by Shelah: http://shelah.logic.at/files/589.pdf In this paper (section 2) he defines the more general notion of the "tree revised power" of two ...

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Various concepts of "closure" or "completion" in mathematics
1 votes

In Model theory we have Skolem hulls: Assuming that each formula has a corresponding Skolem function, one can take a given subset A of a model M, and close it under Skolem functions. This closure Sk(A)...

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