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Results tagged with lo.logic
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user 109573
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.
9
votes
Global Choice bi-interpretable with Global Wellorder?
The answer to the first question is yes: Global Choice is bi-interpretable with Global Well-Ordering.
First, I will prove that a class linear order $(C, <)$ is a well-ordering iff every subset (rather …
- 4,426
3
votes
Surreals and NSA: some foundational issues
Problem 1: There is a definable proper class saturated real-closed field $\mathbb{R}^*$, defined by a slight modification of your and Shelah's construction, such that there is an $\mathrm{OD}_p$ injec …
- 4,426
10
votes
Do the surreal numbers enjoy the transfer principle in ZFC?
A partial answer to the focused question: it's not provable in ZFC that there is an OD class $\mathbb{Z}^*$ such that $(\mathbb{R}, +, \cdot, \mathbb{Z}) \equiv (\mathrm{No}, +, \cdot, \mathbb{Z}^*).$ …
- 4,426
13
votes
Is it consistent with ZFC that the real line is approachable by sets with no accumulation po...
Here's a ZF proof that if $S$ is a chain of sets with $\bigcup S = \mathbb{R},$ then there is $X \in S$ which contains a countable set dense in some nonempty open set.
If there is $X \in S$ such that …
- 4,426
9
votes
Accepted
Must strange sequences wear Russellian socks?
For $n \ge 2$ and $\langle A_i \rangle$ such that $|A_i| = n$ for all $i,$ TFAE:
$|\prod A_i| \neq |\mathbb{R}|.$
$\bigsqcup A_i$ is uncountable.
$\bigsqcup A_i$ is un-orderable.
There are $k \in [2, …
- 4,426
6
votes
Does Well-Ordered Interval Power Set "WOIPS" principle , prove $\sf AC$ in $\sf ZFA$?
I’ll prove the following: over Z set theory, AC is equivalent to cardinal trichotomy holding for the sets between $X$ and $\mathcal{P}^3(X)$ for all infinite $X.$ I suspect the superscript 3 is unnece …
- 4,426
9
votes
Accepted
Building the real from Dedekind finite sets
Q1: There is no such partition. Let $\langle X_n \rangle$ be a countable partition of $\mathbb{R}.$ We will construct $n,$ an open interval $I,$ and an injection $g: \omega \rightarrow I \cap X_n$ wit …
- 4,426
3
votes
Is "the purely probabilistic version of Freiling's axiom of symmetry" disprovable in ZFC?
In any model of ZFC where there is no such set for $c=1$ (e.g., Friedman's model mentioned in Gro-Tsen's answer), there is no $S \subset [0,1]^2$ such that all vertical slices $S_x$ are null and all h …
- 4,426
5
votes
Accepted
Gently changing measure
To question 1, there is such a pair. This is a minor reworking of Ashutosh's example you linked.
Start with $L,$ add an $\omega_1$-sequence of random reals $X=\langle r_{\alpha}: \alpha<\omega_1 \rang …
- 4,426
9
votes
Accepted
The Parity Principle and $\mathbf{C}_2$ (choice for $2$-sets)
Over ZFA, the Parity Principle is strictly weaker. We'll show it follows from Multiple Choice, the assertion that for any family of nonempty sets $\mathcal{F},$ there is $g: \mathcal{F} \rightarrow [\ …
- 4,426
22
votes
What notable theorems cannot be automatically proven without choice using Shoenfield absolut...
$\text{ZF}+ \text{AC}_{\omega}$ is not $\Sigma^1_4$-conservative over ZF and ZF + DC is not $\Sigma^1_4$-conservative over $\text{ZF}+ \text{AC}_{\omega}.$
An example of the former: the sentence $\ome …
- 4,426
12
votes
Accepted
Consistency of a strong Fubini type theorem for measure zero sets
ZFC refutes this principle. Let $\kappa=\text{non}(\mathcal{L}),$ i.e. the least cardinality of a set of reals of positive outer measure. Let $X \subset [0,1]$ be such that $|X|=\kappa$ and $\lambda^* …
- 4,426
5
votes
Accepted
The difference between Baire 2 and 'effectively Baire 2'
It's provable in ZF that every Baire-2 function is effectively Baire-2. It suffices to prove the following:
(ZF) There is an explicit function which maps each Baire-1 function $f: \mathbb{R} \rightarr …
- 4,426
7
votes
Accepted
How hard is it to get "absolutely" no amorphous sets?
Turning my comment into an answer, an $X$ which is the universe of any finitely axiomatized theory with an infinite model must be orderable, and there must be a bijection between between $X$ and $X^2. …
- 4,426
12
votes
Accepted
Is every set being cardinal definable consistent with ZF + negation of Choice?
This is consistent. Kanovei constructed a model $M$ with an infinite Dedekind finite set of reals which is lightface projectively definable. By descending to $L(R),$ we can further assume it satisfies …
- 11.3k