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Questions tagged [axioms]

For questions on axioms, mathematical statements that are accepted as being true, usually without controversy.

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1answer
580 views

Does the set of sets which are elements of every set exist?

In Zermelo-Fraenkel set theory, does the following set exist? $$ A = \{ x \mid \forall y (x \in y) \} $$ I can see why a set which is an element of every set cannot exist (it would break the Axiom of ...
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1answer
28 views

Is there a meaningful analogy between axiomatic systems and vector spaces?

I used to think of axioms within an axiomatic systems as the basis vectors. In this view, every theorem is analogous to a linear combination of the basis vectors, and the proof is analogous to the ...
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1answer
59 views

simplify using laws and axiom of logic [on hold]

$(¬a∨b)∧(a∨b)∧¬a$ So I have been looking at this question all day and and i have no idea how to start. Can someone please help with me proving this algebra logic and what laws I would need to use? ...
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0answers
33 views

Proving the axioms [closed]

Maths is built upon fundamental ideas (such as ‘any two points can be connected with a straight line’). These seem obviously true, but how would you even go about proving something like this?
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4answers
113 views

In Peano's Axioms are the uniqueness of the successor and $x^{\prime}=y^{\prime}\implies{x=y}$ redundant?

In Peano's Axioms are the uniqueness of the successor and the property $x^{\prime}=y^{\prime}\implies{x=y}$ redundant? This seems obvious to me, but I may be missing something. In the various forms ...
0
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3answers
56 views

Name of the basic property of equalities that if a=b then f(a)=f(b).

A basic, fundamental property of equalities is that, if one applies a function on both sides of an equality, the equality still holds. Formally: for any two objects $a$ and $b$ of type $T$ and a ...
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0answers
43 views

Is it possible that $\mathcal{X} = \mathcal{Y}$, yet $\mathcal{X} \in \mathcal{Y}$? [duplicate]

Is it possible for a set to equal another set, yet the former set be an element in the latter set? I.e.: $\mathcal{X} = \mathcal{Y}$, yet $\mathcal{X} \in \mathcal{Y}$
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3answers
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Was there ever an axiom rendered a theorem?

In the history of mathematics, are there notable examples of theorems which have been first considered axioms? Alternatively, was there any statement first considered an axiom that later has been ...
2
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1answer
53 views

Is there a formalization of the link between geometry and analytical geometry?

Geometry and algebra/calculus can be formalized by axioms. Is there a global theory that combines both and establishes correspondences such as the equation of a straight line is $ax+by+c=0$, the ...
0
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1answer
26 views

Does the output of a total function need to be unique against the input?

I'm reading Semantics with Applications and I'm confused about their definition of a total function. They define a function like so: N : Num → Z. And then they ...
4
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2answers
101 views

How are sets defined in reverse mathematics?

Currently going through Simpson's "Subsystems of second order arithmetic", which I believe is the ultimate reference in reverse mathematics, after having completed (more like peeked) Stillwell's "...
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0answers
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Demonstration without the axiom of regularity [duplicate]

Show that for all set $A$ exists a set $B$ such that $B \in P(A)$ and $B \notin A$. I have tried to use the axiom schema of comprehension. I got that there is a set $B$ such that $ x\in B $ iff $x \in ...
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1answer
39 views

Show that the tree $T$ exists by $\Sigma_0^0$ comprehension

Currently I am reading Simpson's Subsystem of Second Order Arithmetic, Chapter IV Weak Konig Lemma. Lemma IV.1.1 (Heine/Borel theorem for $[0,1]$). The following is provable in WKL$_0.$ Given a ...
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1answer
45 views

If something is provable in ZFC, can ZFC prove that ZFC proves it?

IF ZFC proves a Well-formed formula, it means, if $ZFC \vdash \phi$, then, $ZFC \vdash (ZFC \vdash \phi$) ? Generally, if an axiom system proves a wff, the axiom system can prove that it proves ...
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0answers
26 views

Axiomatize inner product spaces in terms of angles?

Inner product space is a vector space $V$ over a field $F$ together with an inner product $P:V\times V\to F$ that satisfies the inner product axioms. This inner product induces an angle $\angle (v,w) ...
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0answers
44 views

Were there any results that were true in old axioms, but are false in modern axioms, and are not obvious paradoxes?

Recently I learned that it can't be proved that mathematical axioms are consistent. And furthermore, in 1900s math was based on an inconsistent system of axioms. So, are there any results, that were ...
7
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4answers
453 views

Why are the axiom of specification is an axiom schema? Why not just a single axiom?

So I was reading about the ZFC axioms, and apparently some of them are actually "axiom schemas." For example, there is the "axiom schema of specification," which basically says that give a set $A$ and ...
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2answers
236 views

Prove cancellation law using peano axioms.

Using Peano axioms, prove $∀x∀y∀z(x+y=x+z→y=z)$. I have been stuck on it for some time, could someone please give a proof? Thanks!
1
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1answer
75 views

What is the point of formalization in mathematics and how does it relate to axiomatization?

Formalization is often presented (1) as a further stage after axiomatization. (It is often said that euclidian geometry was already axiomatized, but not yet formalized ( or not enough), and that the ...
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1answer
36 views

Axiom of countable choice: a question about the domain.

From Wikipedia: The axiom of countable choice or axiom of denumerable choice, denoted $AC\omega$, is an axiom of set theory that states that every countable collection of non-empty sets must have a ...
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2answers
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Some questions about different axiomatic systems for neighbourhoods

I was thinking a few days ago about the development of topology, especially how you arrive at the concept of a topology $\tau$. I knew that a lot of initial ideas came from Hausdorff who defined a ...
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2answers
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Are definitions axioms? [duplicate]

I just want to ask a very elementary question. When we introduce a "definition" in a first order logical system. For example when we say Define: $Empty(x) \iff \not \exists y (y \in x) $ Isn't ...
0
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1answer
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Are axioms in mathematics comparable to hypotheses in experimental sciences? [closed]

Remark: my question deals more particularly with the axioms of set theory, arithmetic, probability theory, etc. I think the status of the axioms in geometry is clearer. The French fictitious ...
2
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2answers
36 views

Vector space basics: scalar times a nonzero vector = zero implies scalar = zero?

I'm working through a linear algebra text, starting right from the axioms. So far I've understood and proved for myself that, for some vector space $V$ over a field $\mathbb{F}$: the additive ...
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1answer
46 views

Why do the hereditarily finite sets model ZF-Infinity, if we need Infinity to talk about them?

As is standard in Kunen, we can talk about $WF$ by working up to it through the "sets of rank $n$" function, $R$, defined in the usual way by iterating collection. In particular, for limit ordinals $\...
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1answer
48 views

ZFC - Prove existence of set {A_1, A_2, A_3, …}

Good morning, I am currently taking a course in axiomatic set theory and I have encountered a problem in showing the existence of a certain infinite (countable) set. Let $A_1$, $A_2$, $A_3$, ... be ...
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1answer
48 views

Are these two extensionality-axioms equivalent?

Let $\epsilon$ be a binary relation on a set $U$. A subset $A \subseteq U$ is called $\epsilon$-transitive iff $$a \mathrel{\epsilon} b \wedge b \in A \Rightarrow a \in A$$ for all $a,b \in U$. For $...
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1answer
41 views

Does the Axiom schema of Replacement imply the Axiom of Infinity? [duplicate]

The axiom of infinity says that the set of natural numbers exists, while the axiom of replacement says that if an object (a member of a set) exists, then all definable mappings of that object yield ...
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3answers
1k views

Why is the Axiom of Infinity necessary?

I am having trouble seeing why the Axiom of Infinity is necessary to construct an infinite set. According to a professor of who's mine teaching a class on "infinity," the Peano axioms are only ...
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1answer
97 views

Why $x=x$? Why $x=y$ and $y=z$ imply $x=z$? (Assume, $x$, $y$ and $z$ are reals.)

I recently have been studying the axiomatic construction of the set of real numbers through the Peano axioms for the natural numbers. It seems to me, the only things needed to proceed with this body ...
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0answers
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Finite axiomatization of second-order NBG

First-order NBG set theory is finitely axiomatizable. The proof of this basically shows that the axiom schemas, in the presence of the other axioms, can be reduced to a finite set of cases, and are ...
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1answer
49 views

Axiom schemas vs second order axioms: which first-order predicates “exist”?

Axiom schemas, such as the PA induction schema, differ from second-order axioms in that they only hold for "definable" predicates, rather for "all" predicates. As a result, you can have non-standard ...
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32 views

Basis of Mathematics

I was reading up on Goedel's Theorems and was wondering what exactly is the basis of Mathematics today and is the discussion around an axiomatic system settled? I know that there are several attempts ...
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1answer
29 views

Verifying axiom of substitution?

In Tao's analysis volume 1, I am introduced to this thing called the axiom of substitution. While constructing real numbers from rationals, he defined reals to be formal limits of Cauchy sequences of ...
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1answer
68 views

Prove that the set of powers of a set, $\{ A^n : n \in \mathbb{N} \}$ exists by using ZFC axioms (without replacement).

I need to prove that the set $\{A^n : n \in \mathbb{N} \}$ using the ZFC axioms (without Replacement). My (rough) plan would be to construct some set containing "more" sets than necessary, then use ...
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3answers
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Why is “points exist” not an axiom in geometry?

I am not sure why "points exist" is not an axiom in geometry, given that the other axioms are likewise primitive and seemingly as obvious.
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1answer
180 views

Axiom of Choice — Why is it an axiom and not a theorem?

My question is only indirectly about the axiom of choice, I just happened to come to the question via the axiom of choice and will use it to illustrate my problem. So far I thought axioms were ...
28
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6answers
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Axiomatic characterization of the rational numbers

We have the well-known Peano axioms for the natural numbers and the real numbers can be characterized by demanding them to be a Dedekind-complete, totally ordered field (or some variation of this). ...
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How to use one set of concrete axioms to build another (smaller) set of abstract axioms.

Say you have a system where your only axioms are: All foo are 10. All bar are 20. All baz are 30. Then from these axioms you can see a pattern, so you say that the underlying axiom is this: All $x$ ...
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25 views

How to determine if an axiom is consistent, independent, complete, and/or categorical

An axiomatic system is: consistent: if no logical contradiction can be derived from the axioms. (Don't see how you can prove there is no logical contradiction possible.) independent: if an axiom ...
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1answer
60 views

How do you show that the three incidence axioms are independent of each other.

How do you show that the three incidence axioms: Incidence Axiom 1. For every pair of distinct points P and Q there is exactly one line l such that P and Q lie on l. Incidence Axiom 2. For every ...
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1answer
80 views

Can we say classical logic has DNE axiom as well because it's equivalent to LEM?

Since according to this page, law of the excluded middle is an axiom of classical logic, Does the paragraph starting with "Classical logic can be characterized by a number of equivalent axioms:" on ...
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1answer
60 views

Are theories in standard mathematics defined on sets or on collections?

Generally, the structures that people study in every-day modern mathematics can be seen as defined on sets, so far as I know. (In the sense that they are collections of objects that don't give rise to ...
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2answers
42 views

tautology vs axiom vs premise

I am trying to sharpen the line among the definitions of tautology, axiom and premise. What I have understood so far is this: Tautology: A statement that is proven to be true without relying on any ...
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1answer
84 views

What's the need for Hilbert's 7th axiom of incidence?

If two planes α, β have a point A in common, then they have at least a second point B in common. I perfectly understand the axiom, but i don't see why it's necessary, and it's kind of ...
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7answers
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In what sense are math axioms true?

Say I am explaining to a kid, $A +B$ is the same as $B+A$ for natural numbers. The kid asks: why? Well, it's an axiom. It's called commutativity (which is not even true for most groups). How do I "...
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1answer
34 views

How to find the additive inverse and additive identity of an element of the tensor-product vector space?

The 8 axioms of holding for a vector space defined in Chapter 5 of the book Robertson's Basic Linear Algebra are easily checked for tensor product expect for existence of (unique?) additive inverse ...
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2answers
39 views

Why do we need the commutative axiom of multiplication?

I think I can prove that $ab = ba$ by other axioms. Am I wrong? Why? Edit: proof updated with notes showing where I actually used the commutative property without noticing. Proof. For any numbers ...
2
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1answer
75 views

$\text {dom}(R)$ and $\text {ran}(R)$ exists for any definition of order pair.

Suppose that we define, for any sets $x,y$, a set $(x,y)$ with the propertie that $$(x,y)=(x',y')\rightarrow x=x'\wedge y=y'(*)$$ Let $R$ be a set. I want to show that the classes $\{x:\exists y((x,y)...
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0answers
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Minimum eligibility to become a theory

Is there any mathematical theory without any axioms? Theories such as set theory, number theory etc., all has axioms in it. I have confusion between mathematical theory and the word theory in usage ...