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-5
votes
1answer
166 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 ...
14
votes
3answers
550 views

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
votes
1answer
392 views

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
votes
1answer
160 views

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 ...
2
votes
1answer
257 views

Consistency of: “The continuum function is injective, and for all infinite cardinals $\kappa$ we have that $2^\kappa$ is weakly inaccessible.”

I asked here about "large powerset axioms" and to my delight, learned that such axioms are being taken seriously. I've been toying with them ever since. My favourite is: "The continuum function is ...
4
votes
1answer
123 views

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) = ...
2
votes
2answers
464 views

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 ...
-1
votes
1answer
188 views

What is an example of a non-axiomatic mathematical system? [closed]

In this wikipedia article on the foundations of mathematics, it says: In practice, most mathematicians ... do not work from axiomatic systems Is this correct? If so, what is an example of this?
3
votes
1answer
425 views

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 ...
5
votes
1answer
181 views

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
2answers
261 views

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
votes
1answer
111 views

what are the Rosser Turquette axioms of Lukasiewicz 3 valued propositional logic?

hi I posted this question on mathematics stackexhange ( http://math.stackexchange.com/questions/468855/what-are-the-rosser-turquette-axioms-of-lukasiewicz-3-valued-propositional-logic ) but did not ...
3
votes
2answers
502 views

Shape of axioms in abstract algebra

When defining abstract algebraic structures (like monoids, groups, etc...), are there some constraints on the shape of the axioms, for the structure to have good properties that we implicitly use in ...
1
vote
1answer
202 views

Infinite board games: sentences about

As a unified approach if we have an ( read any) infinite board game described as $\mathcal{G}$ using a particular axiom set A.. can a sentence be devised in A which automatically answers the basic ...
14
votes
3answers
1k views

Counterintuitive consequences of the Axiom of Determinacy?

I just read Dr Strangechoice's explanation that if all subsets of the real numbers are Lebesgue measurable, then you can partition $2^\omega$ into more than $2^\omega$ many pairwise disjoint nonempty ...
5
votes
8answers
720 views

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
votes
3answers
612 views

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
1answer
317 views

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 ...
12
votes
1answer
926 views

What's the difference between ZFC+Grothendieck, ZFC+inaccessible cardinals and Tarski-Grothendieck set theory?

Say that "U" is the axiom that "For each set x, there exists a Grothendieck universe U such that x $\in$ U", where Grothendieck universes are defined in the usual way (or, if that's unclear, in this ...
4
votes
2answers
454 views

Are there any complete, first-order and unstable theories which have non-categorical second-order formulations?

Since it's not stable, $PA$ fails at being categorical in a power in the worst possible way, having $2^{\lambda}$ models in any uncountable $\lambda$. But $PA$ regains its categoricity in the move to ...
5
votes
0answers
501 views

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 ...
5
votes
1answer
160 views

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 ...
11
votes
5answers
2k views

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 ...
5
votes
1answer
651 views

theorems equivalent to the parallel postulate

Is there a good survey article listing all the theorems of Euclidean geometry that are equivalent to the parallel postulate?
8
votes
1answer
733 views

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 ...
7
votes
2answers
907 views

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
votes
3answers
2k views

Is it possible to show that an infinite set has a countable (infinite) subset, without using the Axiom of Choice?

Let X be an infinite set. Is it possible to show the existence of a countably infinite subset of X without using the Axiom of Choice?
11
votes
5answers
2k views

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 ...
9
votes
2answers
1k views

How would calculus be possible in a finitist axiom system?

I am interested in learning a little more about finitism, currently about which I only know a few encyclopedic paragraphs. I know that during some time, some mathematicians like Kronecker thought ...
14
votes
11answers
3k views

Does the Axiom of Choice (or any other “optional” set theory axiom) have real-world consequences? [closed]

Or another way to put it: Could the axiom of choice, or any other set-theoretic axiom/formulation which we normally think of as undecidable, be somehow empirically testable? If you have a particular ...
1
vote
2answers
689 views

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 ...
3
votes
2answers
1k views

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
votes
3answers
1k views

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 ...
8
votes
5answers
2k views

Axiomatic definition of integers

The real numbers can be axiomatically defined (up to isomorphism) as a Dedekind-complete ordered field. What is a similar standard axiomatic definition of the integer numbers? A commutative ordered ...