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Results tagged with lo.logic
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user 61536
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.
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Is there a formula with one free variable in NBG that defines a class that does not exist?
This question concerns Godel's Theorem on existence of classes in Set Theory of von Neumann–Bernays–Gödel.
This theorem implies that for any formula $\varphi(x)$ with one free variable $x$ whose qua …
- 41.8k
7
votes
1
answer
520
views
The place and year of birth of Henry Maurice Sheffer
I do not know whether this question (in history of math) is proper for MathOverflow, but I know no other places where it can be asked with a hope to obtain an answer.
Reading the biography of Henry Ma …
- 41.8k
2
votes
Accepted
Is $\mathfrak b_a$ a new cardinal characteristic of the continuum?
After thinking some time I realized that $\mathfrak b_a=\mathfrak b$, so this question has answer "No" and this "No" does not help to solve the original question.
To show that $\mathfrak b_a=\mathfra …
- 41.8k
5
votes
1
answer
260
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Is $\mathfrak b_a$ a new cardinal characteristic of the continuum?
By a partial function from $\omega$ to $\omega$ we understand a function $f:dom(f)\to\omega$ defined on an infinite subset of $\omega$.
A family $\mathfrak F$ of partial functions from $\omega$ to $ …
- 41.8k
3
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1
answer
200
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Do the heights of an acute triangle intersect at a single point (in neutral geometry)?
A well-known result of the Euclidean planimetry says that the heights of any triangle have a common point called the orthocentre of the triangle. This result is not true in neutral geometry (i.e., geo …
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2
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Do the heights of an acute triangle intersect at a single point (in neutral geometry)?
I have found a proof (worth a "real bottle of wine") of this fact in Theorem 43.15 on page 430 of the book "Geometry: Euclid and Beyond" of Robin Hartshorne. This theorem is derived from Proposition 4 …
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16
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1
answer
2k
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A contradiction in the Set Theory of von Neumann–Bernays–Gödel?
Thinking on the theory NBG (of von Neumann–Bernays–Gödel) I arrived at the conclusion that it is contradictory using an argument resembling Russell's Paradox. I am sure that I made a mistake in my arg …
- 41.8k
7
votes
1
answer
455
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The existence of definable subsets of finite sets in NBG
This question is motivated by my preceding MO-question on (in)consistency of NBG theory of classes.
Let $\varphi(x,Y,C)$ be a formula of NBG with free parameters $x,Y,C$ and all quantifiers running ov …
- 41.8k
6
votes
0
answers
109
views
Does the Segment-Circle Axiom imply the Circle-Circle Axiom in a non-Euclidean Tarski plane?
By a Tarski plane I understand a mathematical structure $(X,B,\equiv)$ consisting of set $X$, a betweenness relation $B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\times X^2$ satisf …
- 41.8k
4
votes
Accepted
Cofinal monotone maps from $\omega^\omega$ to $\kappa^\kappa$
My former doctoral student Lubomyr Zdomskyy has resolved this problem, noticing that adding $\omega_2$ Cohen reals to a model of GCH produces a model in which the cardinal $\omega_1$ admits a monotone …
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10
votes
1
answer
510
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Cofinal monotone maps from $\omega^\omega$ to $\kappa^\kappa$
Given a cardinal $\kappa$ consider the set $\kappa^\kappa$ of all functions from $\kappa$ to $\kappa$, endowed with the partial order $f\le g$ iff $f(\alpha)\le g(\alpha)$ for all $\alpha\in\kappa$.
…
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6
votes
1
answer
582
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A strong form of the Axiom Schema of Replacement
Let us consider the following stronger version of the Axiom Schema of Replacement (let us call it the Axiom Schema of Replacement for Definable Relations):
Let $\varphi$ be any formula in the language …
- 41.8k
10
votes
1
answer
260
views
The partial preorder on $\mathbb N$ generated by the finite axioms of choice
Let $\mathsf C_n$ denotes the statement:
for any family $\mathcal F$ of $n$-element sets there exists a choice function (i.e., a function $f:\mathcal F\to\bigcup\mathcal F$ such that $f(F)\in F$ for …
- 41.8k
0
votes
1
answer
138
views
Is a right triangle with the given cathetus and the opposite angle constructible in the abso...
Question. Is it possible to construct a right triangle with a given cathetus $a$ and a given opposite angle $\alpha$ using only compass and ruler in the absolute geometry (so without the axiom of par …
- 41.8k
8
votes
0
answers
96
views
Is the hypotenuse operation associative in every Tarski plane?
By a Tarski space I understand a mathematical structure $(X,\mathsf B,\equiv)$ consisting of set $X$, a betweenness relation $\mathsf B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\t …
- 41.8k