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first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
50
votes
0
answers
2k
views
How many algebraic closures can a field have?
Assuming the axiom of choice given a field $F$, there is an algebraic extension $\overline F$ of $F$ which is algebraically closed. Moreover, if $K$ is a different algebraic extension of $F$ which is …
8
votes
1
answer
269
views
What is the consistency strength of "Singular worldly that is inaccessible in an inner model"?
In short, what can we say about the consistency strength of "$\kappa$ is a singular worldly and inaccessible in an inner model"?
Clearly, $0^\#$ exists since we have a singular cardinal which is regul …
20
votes
2
answers
2k
views
What is $\omega_1^{CK}(\mathsf{Ord})$?
We know that if $\alpha<\omega_1^{CK}$ then there is some recursive $R$ such that $(\omega,R)$ has order type $\alpha$.
Let's consider now the ordinals, $\mathsf{Ord}$ with their natural order. This …
53
votes
1
answer
4k
views
When does $A^A=2^A$ without the axiom of choice?
Assuming the axiom of choice the following argument is simple, for infinite $A$ it holds: $$2\lt A\leq2^A\implies 2^A\leq A^A\leq 2^{A\times A}=2^A.$$
However without the axiom of choice this doesn't …
22
votes
3
answers
3k
views
Half Cantor-Bernstein without choice
I had a discussion with one of my teachers the other day, which boiled to the following question:
Assume ZF. Let $A,B$ be sets such that there exist $f\colon A\to B$ which is injective and $g\colon A …
6
votes
0
answers
259
views
Forcing Martin's Axiom without cardinal arithmetic
We know that if $\kappa>2^{\aleph_0}$ and $\kappa^{<\kappa}=\kappa$, then there is a c.c.c. forcing which forces $\sf MA+2^{\aleph_0}=\kappa$. Traditionally, we even start with $\sf GCH$, which simpli …
22
votes
1
answer
1k
views
When will the real numbers be Borel?
In set theory Borel sets are important, but we don't actually care about the sets. We can about the Borel codes. Namely, the algorithm to generate a given Borel set starting with the basic open sets ( …
18
votes
1
answer
4k
views
Countable unions and the axiom of countable choice
Let us denote by ACC the axiom of countable choice, namely the assertion that the product of countably many non-empty sets is non-empty, and denote by UCC the assertion that a countable union of count …
13
votes
1
answer
708
views
Who introduced the notation for $\beth$ numbers and when?
Georg Cantor, when developing the basics of set theory, noted that there are two ways to increase cardinality: power sets and successors (or, in modern terms, the Hartogs operation).1
Eventually the n …
8
votes
1
answer
259
views
Example of a distributive forcing which is entirely improper
One of the standard examples of a forcing which is distributive but not proper (equiv. not closed) is a club shooting into a stationary co-stationary set.
But that forcing is $S$-proper for the statio …
6
votes
5
answers
682
views
Stronger theorem not resulting from proof analysis
Suppose that we proved $\varphi$ from a theory $T$. Often we ask whether or not we could have proved $\varphi$ with a weaker theory, to find out we usually analyze the proof and try to figure out whet …
14
votes
1
answer
799
views
What is the "Prikry–Silver collapse" when CH fails?
We all know and love Cohen reals, and we can (and often do) define the Cohen forcing as partial functions $p\colon\omega\to 2$ with finite domain. The Prikry–Silver forcing is defined as partial funct …
11
votes
1
answer
417
views
Coding the universe into a real over better core models
One of the most incredible results in modern set theory, due to Jensen, is that given any model of $\sf ZFC$, there is a class forcing which adds a real number $r$ and in the extension $V=L[r]$. Moreo …
4
votes
0
answers
174
views
Sequences of sequences of sequences and elementary embeddings
Suppose that $\kappa$ is the critical point of $j\colon V\to M$, and suppose that $\mathcal F=(F_\alpha\mid\alpha\leq\kappa)$ is a sequence such that for every limit ordinal $\alpha$, $F_\alpha$ is a …
10
votes
1
answer
671
views
Is Collection really stronger than Replacement?
The two powerhouse schemata of set theory are Replacement and Collection:
Replacement. For every definable function $f$ and every set $x$, $f"x$ is a set.
Collection. For every definable relation $R$ …