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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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Construction of Proof: Zorn's lemma implies Axiom of choice

I have come across the prove that [Zorn's Lemma ==> AC] but am confused about the central statement, namely that we can take a set of all choice functions on subsets of X (lets just call it X, I ...
CopperCableIsolator's user avatar
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Are $0$-definable sets a countable model of ZF

Above proposition sounds surprising. Can someone check for the mistake? Let me call a formula unary, if it has exactly one free parameter and a set $\emptyset$-definable, if membership is given by a ...
Ascan Heydorn's user avatar
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Since axioms described using normal language, have this ever created problems?

People formalized mathematics using axioms. But axioms still need a natural language to describe them. Have inaccuracy of definitions, caused by using natural language in stating axioms, ever created ...
Den4ik's user avatar
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Are these 3 versions of the Axiom of Infinity equivalent

I am wondering about three variants on the Axiom of Infinity I have come across; in particular, I would like to know if the "weakest" of these could be used to prove the others. The (...
Mark Fischler's user avatar
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2 answers
65 views

Trouble understanding the axiom of foundation

I was reading through the ZF axioms and got to the axiom of foundation which my textbook defined as: $$\forall x(x\neq\emptyset\to\exists y\in x(y\cap x=\emptyset)).$$ Which I found quite confusing. I ...
Reverie's user avatar
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50 views

Is this combination of Lukasiewicz proof systems complete?

I’ve been learning more about axiomatic systems, and somewhat recently came across the following axiomatization for Classical Propositional Calculus a la Lukasiewicz: $(A \to B) \to (B \to C) \to A \...
PW_246's user avatar
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4 votes
1 answer
170 views

Is it weird that my probability theory lecturer thinks that $P(E)=0$ implies that $E=\emptyset$?

I'm an undergraduate student in pure mathematics, and I'm taking a probability theory course based on the book A First Course in Probability by Sheldon Ross. My lecturer defined the conditional ...
Hilbert's user avatar
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Proof that the definition of an n-element set obeys the axiom of extensionality

In Jech's set theory, after defining the unionset axiom, the author defines an n-element set. He first says that $X\cup Y := \bigcup\{X,Y\}$. Then, he makes the definitions $X\cup Y\cup Z := (X\cup Y)\...
Ryan's user avatar
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1 answer
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Why is the unionset axiom framed in the way it is?

I'm currently simulataneously learning set theory from "Notes on set theory" by Yiannis Moschovakis, "Set Theory" by Thomas Jech and "Analysis" by Terence Tao. In two of ...
Ryan's user avatar
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1 answer
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"Irreducible" axiomatic systems?

So I was playing with axiomatic / formal systems and wanted to look into this property about formal systems which could be called "irreducibility". It is similar to independence in that it ...
qucchia's user avatar
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How can something be both a primitive notion and be axiomatically defined like it is here?

This resource states that In mathematics, the notion of a set is a primitive notion. That is, we admit, as a starting point, the existence of certain objects (which we call sets), which we won’t ...
Princess Mia's user avatar
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How does the axiom of specification resolve Russell's paradox?

I've had some trouble understanding exactly how ZFC prevents Russell's paradox and most textbooks I read don't provide a justification for this. Up till now, here's my understanding. Russell's paradox ...
Ryan's user avatar
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1 answer
89 views

Why are all well-defined properties allowed in the Axiom schema of specification in ZFC in FOL?

In first order logic, well-formed formulas are formulas that can be written down using the quantifiers with their bounded variables, free variables, connectives, negation, equality and predicates (in ...
Kevin De Keyser's user avatar
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75 views

Consistency of a set of well-founded sets that is not well-founded

In the proof that $U=V \leftrightarrow Foundation$, the $\rightarrow$ direction can be easily shown by considering $grounded$ sets ($x$ is grounded iff $\forall y: x \in y \rightarrow (y$ is well-...
Niko Gruben's user avatar
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3 answers
86 views

What exactly is the ambient space implicitly referenced in elementary set theory's union and intersection?

After reflecting on the answers in Why does set theoretic union and intersection operate on reverse logic, a follow-up question arose concerning the definition of predicates. In the definition of $A \...
Fomalhaut's user avatar
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2 votes
1 answer
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Peano Arithmetic can prove any finite subset of its axioms is consistent

Timothy Chow writes in a MathOverflow answer [...] here is a classical fact: for any finite subset of the axioms of PA (remember that PA contains an axiom schema and hence has infinitely many axioms),...
Christian Chapman's user avatar
2 votes
1 answer
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Proof within Łukasiewicz's infinite-valued logic $Ł_{א}$

Using the axiom system for Łukasiewicz's infinite-valued logic $Ł_{א}$, I need to construct a proof of the following: ⊢ (A → B) ∨ (B → A) ⊢ (A → (B → C)) → (B → (A → C)) A → B ⊢ (A ∧ C) → B A → B ⊢ ¬B ...
Amilio's user avatar
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0 answers
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How to prove that if two sets belong to the same sets, then they are identical?

The axiom of extensionality states that if two sets have the same members, then they are identical. In symbols, $(\forall x)(\forall y)((\forall z)(z \in x \leftrightarrow z \in y) \rightarrow x=y)$. ...
user107952's user avatar
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4 votes
1 answer
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How can we verify if a structure statisfies ZFC or not?

I'm studying set theory which leads me to some confusions of model theory. For the context, I know that the set $\mathbb{N}$ with the usual operations $(0,+,×)$ is a model of Peano arithmetic. It ...
InTheSearchForKnowledge's user avatar
2 votes
0 answers
22 views

Necessary and sufficient condition for a finite rooted tree to be a redundancy tree

Let $S$ be a consistent finite set of axioms in some first-order language $L$. Recall the definition of an axiom $A$ in $S$ being redundant: $S$ - $\{A\}$ can prove $A$. Now, it can happen that even ...
user107952's user avatar
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1 vote
1 answer
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Justification for the definition of A/R using the power set axiom

I'm currently doing a question from the Foundations of Mathematics written by Kenneth Kunen for self-teaching purposes, and I'm currently stuck on a question that asks me to justify the definition of ...
Syuma's user avatar
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1 answer
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Is there a problem if I don't use $0$ in Peano arithmetic?

Peano arithmetic is the following list of axioms (along with the usual axioms of equality) plus induction schema. $\forall x \ (0 \neq S ( x ))$ $\forall x, y \ (S( x ) = S( y ) \Rightarrow x = y)$ ...
MathMan's user avatar
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3 answers
235 views

Allegedly: the existence of a natural number and successors does not imply, without the Axiom of Infinity, the existence of an infinite set.

The Claim: From a conversation on Twitter, from someone whom I shall keep anonymous (pronouns he/him though), it was claimed: [T]he existence of natural numbers and the fact that given a natural ...
Shaun's user avatar
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2 votes
1 answer
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PA + "(PA + this axiom) is consistent"

By Gödel's second incompleteness theorem, no sufficiently powerful formal system can prove its own consistency. I was wondering what happens if one tries to manually append an axiom stating a formal ...
volcanrb's user avatar
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5 votes
1 answer
192 views

Recursive definition of union seemingly informal?

In my set theory course, we introduced the Axiom of Union. For any set $x$ there exists a set $y$ s.t. all elements of $y$ are elements of some $z\in x$. Notation for this set is $y= \bigcup x$. $\...
Amitai's user avatar
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4 votes
3 answers
442 views

One more time, ZF based proof that set of all sets does not exist

I know it has been asked several times, but there is always one step that I don't see. My argumentation goes as it follows: Given the set $\mathbb{V}$ set of all sets, then $V \in V$. Because of the ...
pdaranda661's user avatar
2 votes
0 answers
129 views

Why are the probability/Kolmogorov axioms called "axioms"?

Question I don't understand why the three axioms of probability (Kolmogorov axioms) are treated as axioms, not as definitions. Perhaps it hints at a larger problem: I don't understand why any axiom is ...
Cynicrom's user avatar
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1 answer
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Prove that the union of all lines containing a point A is the plane.

I need to prove this with the knowledge of incidence and order axioms. Let $X$ be a set with all the points forming all the lines containing the point $A$, and let $Y$ the set of all points in the ...
LightL96's user avatar
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1 answer
95 views

In the axiomatic treatment of natural numbers, can we define what a natural number is?

In his book Number Systems and the Foundations of Analysis, Elliot Mendelson first defines a Peano system (p. 53). He then goes on to prove that two arbitrary Peano systems are isomorphic — they are ...
Mostafizur Rahman's user avatar
1 vote
1 answer
107 views

Can this presentation of reflection be considered foundational?

Working in the first order language of set theory. By $R$-bounded quantifiers its meant those of the form $\forall x \ R \ a \, ( \cdots) $ , or $ \exists x \ R \ a \, ( \cdots) $, and these are ...
Zuhair's user avatar
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1 vote
1 answer
68 views

Understanding the axioms for neighbourhoods and their independence/consistency

After reading answers to this question I can't help being confused about axioms 2 and 4 being truly independent, or about their true meaning for that matter. Recalling them, for a given set $X$: If $...
AmazingWouldBeGreatBut's user avatar
1 vote
1 answer
119 views

Is the Axiom of Completeness logically equivalent to "There is no proper superset of $\mathbb R$ that is an ordered Archimedean field"?

The Axiom of Completeness can be formulated as: There exists a set $R$ such that: $R$ is an ordered Archimedean field Any nonempty subset of $R$ with an upper bound has a least upper bound. Recently,...
SRobertJames's user avatar
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0 votes
1 answer
68 views

Is consistency sufficient for existence?

In his Mathematical Analysis I, Zorich says the following after introducing the reals axiomatically: In relation to any abstract system of axioms, at least two questions arise immedi- ately. First, ...
EE18's user avatar
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0 votes
2 answers
81 views

Can I combine axioms to have less properties to verify independently? [closed]

For example, given that a linear function is defined by its satisfying the properties $f(x+y)=f(x)+f(y)$ and $f(ax)=af(x)$, would it be okay to only check $f(ax+by)=af(x)+bf(y)$, or maybe even $f\left(...
AnotherSherlock's user avatar
0 votes
1 answer
32 views

Need hints (advice) to prove $(\forall a,b,c,d \in \mathbb{R}) (a < b \wedge c<d) \Rightarrow ad+bc < ac +bd$

I'm trying to prove this ( source : my uni's textbook says that it's trivial). $$(\forall a,b,c,d \in \mathbb{R}) (a < b \wedge c<d) \Rightarrow ad+bc < ac +bd$$ So far, I've managed to get ...
runtotherescue's user avatar
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0 answers
51 views

Don't we need an axiom for $\omega_1$? [duplicate]

As we needed an axiom to take the supremum of the finite ordinals, how are we allowed without another axiom to take the supremum of the countable ordinals?
Nathan Kaufmann's user avatar
2 votes
0 answers
31 views

Use closure under addition of positive numbers to prove that a<b => a+c < b+c

I'm starting R.P Burn "Number and Functions, Steps into Analysis". In Chapter 2, the basic properties of inequalities are derived, and question 8 asks to prove using the fact that positive ...
user2958738's user avatar
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0 answers
44 views

Need help showing that associativity works in a set of two symbols

Let $ \mathbb{P} = \left \{ \bullet , \blacksquare \right \}$ be a set where elements $ \bullet \neq \blacksquare$. For operation of addition in $\mathbb{P}$ we define: $$\bullet + \bullet = \...
runtotherescue's user avatar
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0 answers
83 views

Why does the class axiom suffice?

I am reading a book called ‘A Book of Set Theory’, by Charles C. Pinter. Quite early on he mentions the class axiom, which goes like ‘If S(x) is any statement about an object x, there exists a class ...
Максим Неважно's user avatar
13 votes
3 answers
1k views

Is the Axiom of Choice inconsistent with Countable Additivity?

Consider a fair lottery among a countably infinite number of people. The Axiom of Countable Additivity says this is impossible to construct: If all people have a positive (and equal) probability of ...
Joseph Camacho's user avatar
0 votes
1 answer
239 views

Precise axiomatic definition for the equality "=" as a binary relation

Question: What is a simple yet precise definition for "=" as a binary relation? My try: I find two definitions for "equality relation" which seems to be contradictory. The first ...
dodo's user avatar
  • 788
10 votes
2 answers
1k views

If set theory only contains the notions of “set” and “is a member of” as primitives, how can an axiom of set theory refer to a “formula”?

It's said that the primitive concepts of set theory are those of "set" and "membership", then all axioms of set theory must begin with "Let $A$ be a set" or "Let $x\...
RataMágica's user avatar
8 votes
2 answers
429 views

Is axiom of replacement nicely stateable in the language of ETCS?

ETCS has a nice category-theoretic formulation: "well-pointed topos with a natural numbers object and axiom of choice." I'm too new to topoi to really understand all of what's going on, but ...
Cobalt _000's user avatar
-2 votes
2 answers
179 views

Defining Division by Zero as an Axiom

It is an accepted knowledge in mathematics that division by zero is undefined. But this is not intuitive to me. I'm not a mathematician, but how about if there is an axiomatic definition that any ...
Noble_Bright_Life's user avatar
1 vote
0 answers
68 views

Axioms for equality [duplicate]

For first order logic without equality, what are the exact axioms we give to define the relation of equality ? I can't find the exact axioms even in Wikipedia... Any reference to any article or a list ...
user avatar
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2 answers
94 views

Prove that if $x\not=y,$ then $\{a\}\not=\{x,y\}$ for any sets $a$ only using three given axioms

Prove that if $x \not = y$ , then $\{ a \} \not = \{ x , y \}$ for any set $a$. It is a trivial problem, but I wanted to make my life more difficult and prove it using only three axioms if possible : ...
Alisher's user avatar
1 vote
0 answers
73 views

At an intuition/axiom level, what are the differences between algebra and geometry?

This is a [potentially] soft question that I feel like encompasses some pedagogical elements but possibly also advanced foundations, so I hope this is the proper forum to ask. I'll give some ...
zhuli's user avatar
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3 votes
1 answer
82 views

Do you need axiom of choice to proof $2^\mathbb{N}$ is nonempty? [duplicate]

Define $2^\mathbb{N}$ as the set $$ 2^\mathbb{N} = \prod_{i \in \mathbb{N}} \{0,1\} = \{ f: \mathbb{N} \to \{0,1\} \}.$$ It seems we do not need the axiom of choice to show that this set is non-empty; ...
Inzinity's user avatar
  • 1,763
2 votes
1 answer
80 views

Can cardinality be defined in ZF-Regularity without the axiom of choice and without Scott's trick?

In ZF-Regularity, can we define a notion of cardinality such that for all sets $A$ and $B$, $card(A) = card(B)$ iff there exists a bijection from $A$ to $B$? If we add a function symbol C to the ...
Hussein Aiman's user avatar
0 votes
0 answers
33 views

A countable union of countable sets is countable requires the axiom of choice [duplicate]

By countable sets, I refer to countably infinite sets. We can very well prove that the natrual numbers and $\mathbb{N} \times \mathbb{N}$ have the same cardinality, for a bijection take the Cantor ...
Shthephathord23's user avatar

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