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Results tagged with lo.logic
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user 4213
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.
5
votes
Accepted
Exponent function as uninterpreted function in first order logic
The function $f(m,n)=m^n$ is primitive recursive, so expressible in
first-order arithmetic: there is a formula in three free variables
$F(m,n,p)$ over the language of first-order arithmetic
which is v …
- 20.8k
2
votes
Accepted
Natural number properties as uninterpreted functions in first order logic
If $f(n)$ is a predicate in the first order language of arithmetic,
then "there are infinitely many $n$ such that $f(n)$ holds" can be
expressed as
$$\forall m\in\mathbb{N}\ \exists k\in\mathbb{N}:f(m …
- 20.8k
16
votes
Accepted
Order types of positive reals
Yes, one can have any countable ordering. Indeed any countable totally
ordered set can be embedded in $\mathbb{Q}$. Write your ordered set as
$ \lbrace a_1,a_2,\ldots \rbrace $
and define the embeddin …
- 20.8k
2
votes
Rank of a free module without the axiom of choice
Let $A$ and $B$ be infinite sets. Let $M$ be a rank $|B|$ module
with basis $e_b$ for $b\in B$. If we take $|A|$ elements $m_a$ of $M$, then each can
be expressed in terms of finitely many of the $e_b …
- 20.8k
10
votes
11
votes
How do they verify a verifier of formalized proofs?
Arguably Norman Megill's Metamath
http://metamath.org/
is not a mainstream proof verifier. But various third parties
have written short programs to check his deductions, starting
with Ralph Lieven's m …
- 20.8k