Questions tagged [axioms]
The axioms tag has no usage guidance.
131
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Relationship between computational undecidability and axiomatic undecidability
On surface, these seem two completely different class of problems. One class represent statements which can't be proved or disproved in an axiomatic theory. For example
One can write down a ...
1
vote
1
answer
509
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Pasch axiom and Pythagorean field condition?
I am looking for a reference for the claim that the Pasch axiom is equivalent to the Pythagorean field condition, and with respect to what base theory this should be true.
Since posting the question, ...
1
vote
0
answers
296
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Can we strengthen the axiom of choice to settle the generalized continuum problem?
By the generalized continuum problem, I mean the following: given an infinite cardinal $\kappa$, find the order type of the set of all cardinals strictly between $\kappa$ and $2^\kappa$.
Now whenever ...
13
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0
answers
852
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Arguments against Freiling's argument against Continuum Hypothesis
Freiling's axiom of symmetry ($\sf AS$) is known as a justification for falsity of Continuum Hypothesis. Freiling in his 1986 paper, Axioms of symmetry: throwing darts at the real number line, ...
49
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4
answers
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How undecidable is the spectral gap?
Nature just published a paper by Cubitt, Perez-Garcia and Wolf titled Undecidability of the Spectral Gap, there is an extended version on arxiv which is 146 pages long. Here is from the abstract:"Many ...
2
votes
0
answers
111
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Quasigroups extracted from the rational numbers and division
Consider a quasigroup $(Q,/)$, that is, Q is a set and for $\forall a,b\in Q$ there are unique solutions to the equations $x/a=b$ and $a/y=b$. How to find a maximal set of independent representants of ...
10
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1
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884
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Is the axiom schema of replacement used in algebraic number theory (or more generally outside logic)
Here's a precise question. Does Wiles' proof of FLT run just fine in the set theory that logicians would perhaps call "Zermelo + choice" -- i.e. drop the axiom schema of replacement but assume the ...
4
votes
1
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167
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Does this axiom (a weak form of class valued choice) has a name?
At some point in my work (which has nothing to do with set theoretics foundation) I need to consider the following axiom:
For any set $X$, any class $V$ with a surjective map $f : V \...
6
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0
answers
204
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A generalization of SOCA
Roughly speaking, SOCA (Semi Open Coloring Axiom) says that for an open coloring of the unordered pairs over an uncountable separable metric space you can always find an uncountable homogeneous subset ...
-3
votes
1
answer
255
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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 ...
20
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3
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Characterize the category of rings
(Sub)categories of many well-studied mathematical objects have been characterized purely in terms of their morphisms. Some (famous) examples:
Sets and functions, due to Lawvere.
Modules over some ...
12
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1
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803
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Transfinitely extending $\sf PA$ — can we get stronger than $\sf ZFC$?
Let $\sf PA$ denote the theory of natural numbers with constants $(0, 1)$ and binary operators $(+,\times)$ based on the first-order predicate calculus with equality, having the following axioms, ...
0
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1
answer
262
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Possible no standard use of replacement axiom
The idea is to build in ZFC using replacement, a set REPLACEMENT(x ∈ A: TERM(x))
from a set and a term in the same way the set {x ∈ A: FORMULA(x)} is built
using specification from a set and a formula....
4
votes
1
answer
183
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Class theory with support for self-application of class functions?
To every natural number $n$, we can assign its Church numeral $\underline{n}.$ A formal definition would be:
$\underline{0}(f)=\mathrm{id}_{\mathrm{dom}(f)}$
$\underline{n+1}(f) = \underline{n}...
2
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2
answers
555
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Have axioms / axiom schemata of this flavour been proposed or otherwise considered?
With the exception of a few miscellaneous cases, the axioms (and/or schemeta) of ZFC can roughly be divided into two kinds:
Those that guarantee the existence of more complicated sets, given that ...
3
votes
1
answer
952
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Please recommend a nice and concise math book on probability theory [closed]
My intention is neither to learn basic probability concepts, nor to learn applications of the theory. My background is at the graduate level of having completed all engineering courses in probability/...
4
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1
answer
233
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Order Types and Replacement Schema
Using Replacement Schema we can prove any well-ordering is isomorphic to an ordinal number.
Q: Is the following consistent?
$ZFC-Rep+\neg Rep+\text{Any well-ordering is isomorphic to an ordinal ...
4
votes
2
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434
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On wild behavior of $\omega_{1}$ in the absence of some essential axioms of $ZFC$
The regularity of $\omega_{1}$ is one of the most well known facts of set theory. But it seems that in order to prove this simple fact we need the "full power" of mathematics! For example by an ...
2
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1
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231
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what are the Rosser Turquette axioms of Lukasiewicz 3 valued propositional logic?
hi I posted this question on mathematics stackexhange ( https://math.stackexchange.com/questions/468855/what-are-the-rosser-turquette-axioms-of-lukasiewicz-3-valued-propositional-logic ) but did not ...
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5
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Minimal subset of axioms for ZFC
Hello all, one may look for "minimal system of axioms" for ZFC (or any other
theory) in the following (unusual) sense : say that a subset S of ZFC is
"sufficient" if there is an explicit procedure ...
7
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8
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Result that follows from ZFC and not ZF but are strictly weaker than choice
A number of results that people use that require the axiom of choice (i.e. do not follow from ZF alone) are known to actually imply the axiom of choice. Therefore, one might naturally wonder whether ...
4
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3
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Survey of finite axiomatizability for relational theories?
An $L$-theory $T$ is finitely axiomatizable if there is a finite set $A$ of $L$-sentences with the same consequences as $T$, i.e. such that $M \models T$ iff $M \models A$ for every $L$-structure $M$. ...
6
votes
1
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492
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What is known about size-restricted power set axioms?
What is known about ZF without powerset but with an axiom "every set
has a set of all its countable subsets"?
This seems stronger than positing that the set of natural numbers has
a powerset, though ...
9
votes
1
answer
830
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Axiom of class collection
One version of the Axiom of Collection says that any surjection $A\to B$ from a class $A$ to a set $B$ is factored through by some surjection $C\to B$ where $C$ is a set.
Note that assuming $B$ is a ...
5
votes
0
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740
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two versions of the nested interval property
There appear to be two different nested interval properties for the reals with the punchline "... then the intersection of the intervals is non-empty", and I'd like to know their respective histories (...
4
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1
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184
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Theories and indiscernible propositions
Are there known examples of statements which are strong from a proof-theoretic standpoint but which are indistinguishable by one set of axioms (or proof system) yet distinct according to a stronger ...
15
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5
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6k
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getting rid of existential quantifiers
It seems to me that for most of the twentieth century, axiomatic foundations for mathematical theories were constructed with the (mostly allied) goals of minimizing the number of primitive notions and ...
7
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2
answers
1k
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Are there uncountably many essentially inequivalent versions of Mathematics?
Hi everyone,
Disclaimer 1: logic and set theory are a long way from my field, so apologies in advance if I demonstrate extreme ignorance or stupidity, and please correct me if (when?) I write stupid ...
6
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2
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2k
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Axiom of Computable Choice versus Axiom of Choice
What would be the consequence of requiring that any choice function be computable; i.e. using as the foundational basis ZF + ACC? Does it make a difference if we admit definable functions?
I guess I ...
15
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3
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1k
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Any paradoxical theorems arising from large cardinal axioms?
If we accept the axiom of choice, we take the responsibility of living in a world in which, e.g., a ball in Euclidean 3-space is equiscindable to two isometric copies of itself (Banach-Tarski). So we ...
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2
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958
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What is cardinality of the set of true undecidable minimal sentences in a formal theory of aritmetic
Let T be a true theory of arithmetic to which the incompleteness theorems apply. Consider two sentences in the language of T to be equivalent if they are provably equivalent over T. How many ...