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Results tagged with lo.logic
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user 6794
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
Accepted
Large almost disjoint family on $\mathbb{N}$ with property $\mathbf{B}$
One of the standard examples of an almost disjoint family of cardinality $\mathfrak c$ is the set of paths through the complete binary tree $2^{<\omega}$ (identified with $\omega$ via your favorite bi …
- 73.2k
8
votes
Accepted
Heuristic interpretations of the PA-unprovability of Goodstein's Theorem
You've shown how to prove, in PA, the statement "the Goodstein sequence starting with $p$ terminates" for any given $p$. But once $p$ is given, that statement has a proof in PA that just consists of c …
- 73.2k
10
votes
Accepted
Additive, multiplicative, and Dedekind infiniteness in ${\sf (ZF)}$
A theorem of Tarski says that the statement "all infinite sets are multiplicatively infinite" implies the axiom of choice (AC). But a theorem of Sageev says that "all infinite sets are additively infi …
- 73.2k
6
votes
Jensen's proof that $\diamondsuit$ holds at subtle cardinals
The "Suppose not" in the second paragraph is the reason for the fact in the third paragraph (and is thus what leads to the desired contradiction).
The point is that the definition of $\langle S_\alpha …
- 73.2k
10
votes
Why is it OK to rely on the Fundamental Theorem of Arithmetic when using Gödel numbering?
Stefan has explained the essential points well, but let me add some details and a reference. For the first incompleteness theorem, one can work with a very weak theory of arithmetic. One needs to be …
- 4,713
12
votes
What would you do with a new model of linear logic?
The first thing I'd do with a new model is to see how it relates to things I already know. For example, how does it relate to game semantics?
What happens if I add weakening and contraction to the log …
- 73.2k
178
votes
Set theories without "junk" theorems?
I apologize for posting as an answer what should really be a comment, connected to one of Jacques Carette's comments on my earlier answer. Unfortunately, this is way too long for a comment. Jacques …
- 6,403
7
votes
Accepted
"Compactness length" of Baire space
Baire space is the union of $\mathfrak d$ (the dominating number) compact subsets. So, using equivalence relations that collapse those sets one at a time (i.e., one equivalence class is the set to be …
- 73.2k
7
votes
Accepted
Impredicativity, definition, recursion and conservatism
The formula $Gx\leftrightarrow A(G,x)$, expressing that $G$ is a fixed-point of the operator defined by $A$, is not sufficient, by itself, to uniquely characterize $G$. That operator may have many fix …
- 73.2k
16
votes
Ultraproducts of Banach spaces versus model theoretic ultraproduct
As a logician, I take the model-theoretic notion of ultraproduct as the primary one, so the following formal connection describes how to get the Banach-space ultraproduct from the model-theoretic one. …
- 73.2k
11
votes
Does every consistent extension of ZF have a model in the minimal transitive model of ZFC?
As Joel said, most of the consistent extensions $T$ of ZFC are not in the minimal transitive model $M$ of ZFC and therefore don't have models in $M$. It seems worth noting that this is the only reason …
- 11.3k
7
votes
Accepted
Minimum number of dense sets to make a filter an ultrafilter
No; $\mathfrak u'=\mathfrak c$.
To prove it, consider any $\mathcal C\subseteq[\omega]^\omega$ with cardinality $<\mathfrak c$. Working modulo finoite subsets of $\omega$ , and closing under (finitary …
- 73.2k
5
votes
Accepted
Decomposition of an ultrafilter on the fibers of a map
First, let me dispose of the trivial cases where $f$ is constant or one-to-one on a set in $\mu$. In the case of constant $f$, say with value $i$, you can take $\eta$ principal at $i$ and let $\mu_i$ …
- 73.2k
26
votes
Accepted
Why is there a need for ordinal analysis?
The axioms of first-order arithmetic include the induction schema, which says that, for every formula $A(x)$ with free variable $x$, the conjunction of $A(0)$ and $\forall x\,(A(x)\rightarrow A(x+1))$ …
- 2,031
34
votes
Accepted
Interpretation of the Second Incompleteness Theorem
For the philosophical point encapsulated in (*) in the question, it seems that corollaries of the second incompleteness theorem are more relevant than the theorem itself. If we had doubts about the c …
- 17k