# 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 ...
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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 ...
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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 ...
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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 (...
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### 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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### 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 ...
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### 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-...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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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 = \... 0 votes 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 ... 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 ... • 1,617 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 ... • 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\... • 316 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 ... • 135 -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 ... 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 ... 0 votes 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 : ... 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 ... • 2,571 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; ...
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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 ...
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 ...