All Questions
6,026 questions
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Can this form of reflection be consistent?
Is this form of reflection consistent?
First I'll begin by clarifying the notation I'm using here:
By a quantifier being relativized or bounded it means that the first occurrence of the quantified ...
- 11.3k
-3
votes
1
answer
189
views
Propositional logic without rules of inference and assumptions (except MP) [closed]
I was wondering whether it would be possible to do propositional logic without any rules of inference and assumptions (except modus ponens).
I have the following axioms:
$ p \to (q \to p) $
$ (p \to (...
-3
votes
1
answer
101
views
Order-embeddability of ${\frak b}$ and ${\frak d}$ in $\mathbb{R}$ [duplicate]
The starting point of this question is the observation that in ${\sf (ZFC)}$, all ordinals $\alpha < \omega_1$ can be order-embedded in $\mathbb{R}$.
Let $\omega^\omega$ denote the set of all ...
- 48.9k
-3
votes
1
answer
234
views
A common name for a functorial construction of Commutative Algebra?
I am interested whether the following construction naturally appearing in Commutative Algebra has some know and acceped name.
Given a commutative monoid $(M,+)$ and a set $X$, consider the family $F(...
- 41.9k
-3
votes
1
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341
views
What is the intuitive notion that ZF-Extensionality-Foundation+Collection can be said to capture? [closed]
This question has been moved to philosophy.stackexchange.com
I'll try to abbreviate it here: the question asks about the "informal notion" that the fragment of $\text{ZFC}$ that is axiomatized by ...
- 11.3k
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1
answer
330
views
Is there a precise definition of "mathematical formula"? [closed]
In the Wikipedia article for Formula (which has no references), it is claimed that:
"The informal use of the term formula in science refers to the general construct of a relationship between given ...
- 3,359
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1
answer
262
views
An axiomatic system with a set of constants that form a complete ordered field [closed]
I am developing a ZFC axiomatic system where together with the empty set, there is a singular (and huge) set of constants that are themselves sets and form a complete ordered field (cof) these ...
- 201
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3
answers
2k
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Infinite CPU clock rate and hotel Hilbert [closed]
Suppose we have a computer with infinte CPU clock rate, infinite CPU registers, storage etc. Lets run a program that could look something like this:
A=1
while A>0
A = A+1
repeat
We start the program ...
- 49
-4
votes
1
answer
224
views
Do monoid homomorphisms from $X^X$ to a group factor through $\text{Sym}(X)$? [closed]
Let $X$ be a set and let $(X^X,\circ)$ denote the monoid of all maps $f: X\to X$, together with composition. Let $(\text{Sym}(X),\circ)$ be the group of all bijections from $X$ to itself.
Does there ...
- 48.9k
-4
votes
2
answers
399
views
Two equivalent statements about formulas projected onto an Ultrafilter
Question 1:
In the same language, let $ X $ be a nonempty set, and let $ \{ (\forall x_{x(i)} f(i)) \ | \ i \in X \} $ be a set of formulas. We use $ x(i) $ to denote the index of the variable on ...
- 127
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votes
2
answers
224
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What happens if I replace a *unique natural number* that form a commutative Monoid with *the set of integers* Z that form a commutative Ring? [closed]
In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was ...
-4
votes
1
answer
267
views
Is Nested Selection equivalent to AC?
Nested Selection: For every infinite set $G$ of pairwise disjoint infinite sets such that any two distinct elements $x,y$ of $G$ either "$y$ is a set of proper supersets of elements of $x$ and ...
- 11.3k
-4
votes
1
answer
173
views
To which arithmetic\set theory this theory is bi-interpretable?
$\textbf{Logic:}$ Mono-sorted first order logic with equality.
$\textbf{Extralogical Primitives: } <, \in$
$ \textbf{Axioms:}$
$ \textbf{Order:} \ x < y < z \to x < z $
$ \textbf{...
- 11.3k
-4
votes
1
answer
203
views
Is there a procedure to derive models from axiomatic systems? [closed]
Is there a systematic procedure to construct a model of an axiomatic system from the system itself?
For example given the abstract postulates of a ring we can show that the integers satisfies them ...
- 4,862
-4
votes
0
answers
133
views
Which arithmetic\set theory is synonymous with this theory?
$\textbf{Logic:}$ Mono-sorted first order logic with equality.
$\textbf{Extralogical Primitives: } <, \in$
Define: $x > y \iff y < x$
Define: $x \leq y \iff x < y \lor x=y$
$ \textbf{...
- 11.3k
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votes
0
answers
189
views
Can ZFC be interpreted in this infinitary logic theory?
Working in language $\mathcal L_{\Omega^+,\Omega^+}$ where $\Omega$ is the first strongly inaccessible cardinal. If we add a primitive partial $\Omega$-ary function $F$ and a primitive constant $\...
- 11.3k
-4
votes
1
answer
140
views
About the definitions of well-foundedness in this extension of NFU that interprets ZFC?
Lets see how the world of sets could look like from the perspective of $\sf NFU$. So, here we work within the first order language of set theory, with the following extra-logical axioms:
1. Quine atom:...
- 11.3k
-4
votes
1
answer
198
views
Is Bounding Reflection consistent?
Working in the first order language of set theory.
Let $\varphi^{*B}$ be the formula obtained from $\varphi$ by merely bounding all open quantifiers in $\varphi$ by the symbol "$B$".
Here a ...
- 11.3k
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votes
1
answer
3k
views
Kadison-Singer problem
The Kadison-Singer problem is the following statement:
for any $\epsilon >0$, there exists $r\in \mathbb N$ such that
for any bounded operator $A$ on $\ell^2(\mathbb Z)$, there exists a partition $(...
- 16.2k
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votes
1
answer
233
views
First research papers in mathematical logic [closed]
Hello I'm a software engineer who just wants to start research in mathematics soon. I'm interested in the foundations and hence I'm picking mathematical logic. As I have never touched undergraduate-...
-5
votes
1
answer
231
views
Are equinumerous size preserving models of a theory isomorphic?
If by a size preserving model we mean any bijection between any two elements of it is an element of it. Then:
is it a thoerem of $\sf ZFC$ that for any theory $T$ any two equinumerous size preserving ...
- 11.3k
-5
votes
1
answer
390
views
Can we blend ZFC with true arithmetic?
Can we have a consistent theory whose signature is $(=,\in, S, +, \times)$ standing for identity and membership binary relations and the successor total unary function, addition and multiplication ...
- 11.3k
-5
votes
0
answers
250
views
Can Cardinality Theory capture ZFC?
Cardinality Theory "CT" is a theory of sets of cardinals and links between them, only sets of cardinals can be assigned cardinalities. The links are unordered edges linking cardinals, they ...
- 11.3k
-6
votes
1
answer
295
views
Is the least ordinal containing all countable ordinals defined by a formula an element of itself? (Out of date, see below)
[UPDATE] The question has been updated and is mostly unrelated to the question above the line. The updated question is below the line.
The following argument supports a "yes" answer; is it convincing?...
-6
votes
1
answer
779
views
De-Lifting Lemma, does it hold? [closed]
Let $\sigma$ denote an independent simultaneous substitution. Now I wonder if the following holds:
If $\Gamma \vartriangleright (A\ (\sigma\ \tau))\ \rho$ then there are $\psi$, $\phi$ such that $\...
Countably Infinite
-8
votes
1
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351
views
Are there overwhelmingly more finite monoids than finite spaces? [closed]
A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...
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