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Results tagged with lo.logic
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user 2000
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.
80
votes
Accepted
What is the high-concept explanation on why real numbers are useful in number theory?
The Gödel Speedup Theorem provides some explanation why real numbers (and variants) are useful in proving statements in number theory.
Real numbers, complex numbers, and $p$-adic numbers are second-o …
71
votes
Accepted
How many orders of infinity are there?
For asymptotic domination, commonly denoted ${\leq^*}$ and often called eventual domination, this has been answered by Stephen Hechler, On the existence of certain cofinal subsets of ${}^{\omega }\ome …
- 44.4k
49
votes
Accepted
In what rigorous sense are Sperner's Lemma and the Brouwer Fixed Point Theorem equivalent?
Sperner's lemma is not equivalent to Brouwer's Fixed Point Theorem. All that one can prove directly from Sperner's Lemma is the following weaker statement.
Approximate Fixed Point Theorem. Let $K$ be …
- 44.4k
42
votes
Accepted
Is the Riemann Hypothesis equivalent to a $\Pi_1$ sentence?
I don't know the best way to express RH inside PA, but the following inequality
$$\sum_{d \mid n} d \leq H_n + \exp(H_n)\log(H_n),$$
where $H_n = 1+1/2+\cdots+1/n$ is the $n$-th harmonic number, is k …
- 44.4k
42
votes
Accepted
"Transitivity" of the Stone-Cech compactification
The answer to Q1 is no. This has been well studied in set theory; you're basically asking whether any two non-principal ultrafilters on $\mathbb{N}$ are comparable under the Rudin-Keisler ordering. Va …
- 44.4k
41
votes
Who needs Replacement anyway?
I think the main reason replacement is seen as an essential part of ZF is that it naturally follows from the ontology of set theory, as do the other axioms of ZF. The ontology of set theory is rooted …
- 44.4k
39
votes
Why is this new result such a big deal?
The statement in question, frequently denoted $\mathsf{RT}^2_2$ in the context of reverse mathematics, is the instance of the infinite Ramsey theorem for unordered pairs and two colors. Specifically, …
- 44.4k
39
votes
Most 'unintuitive' application of the Axiom of Choice?
I highly recommend reading this paper by Chris Hardin and Al Taylor, A Peculiar Connection Between the Axiom of Choice and Predicting the Future (Wayback Machine), as well as this shorter piece by Mik …
36
votes
Accepted
Propositional Logic, First-Order Logic, and Higher-Order Logics
This is a long list of questions! These are all related to a certain extent, but you might consider breaking it up into separate questions next time.
Proof theorists tend to prefer systems with many …
- 44.4k
35
votes
Accepted
Internal logic of the topos of simplicial sets
Short answer: the Kreisel-Putnam axiom $(\lnot p \to (q \lor r)) \to ((\lnot p \to q) \lor (\lnot p \to r))$ is not an intuitionistic tautology but it is valid for any subobjects of an object in the t …
- 44.4k
33
votes
Accepted
Does $\operatorname{Con}\sf(ZF)$ imply $\operatorname{Con}\sf(ZF + \operatorname{Aut}{\bf C ...
The use of inaccessible cardinals is not necessary here, the Baire property works just as well as Lebesgue measure. Shelah (Can you take Solovay's inaccessible away, Isr. J. Math. 48, 1984, 1-47) show …
- 44.4k
32
votes
Accepted
Forcing as a new chapter of Galois Theory?
Yes and No... There are strong parallels between forcing and symmetric extensions and field extensions and this way of thinking has been fruitful. However, like in the case of general ring extensions …
- 44.4k
31
votes
Accepted
Using consistency to create new axioms in set theory
Such constructions are interesting! However, they are often done with PA instead of ZFC (see note). For an interesting discussion, I recommend Torkel Franzén's book Inexhaustibility: a non-exhaustive …
- 44.4k
31
votes
Has decidability got something to do with primes?
The role of primes in Gödel's Incompleteness Theorem can be better understood by looking at Robinson's Q, which is one of the weakest theories of arithmetic for which Gödel's Incompleteness Theorem ho …
- 44.4k
30
votes
Reductio ad absurdum or the contrapositive?
Strictly speaking, the contrapositive of a statement which is not an implication doesn't make sense. However you can always fake the implication, the contrapositive of $\top \to A$ (or just $A$) is $\ …
- 44.4k