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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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### Show that $x = \{\{x\}\}$ is not a set using the axioms of $ZFC$ [duplicate]

I am currently studying a course on Set Theory with the axioms of $ZFC$ and I understand that the axiom of foundation/regularity prevents self membership, i.e. $x \in x$ is not allowed. My question is:...
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### Relation between Axiom of Foundation and $\in$-induction

At the end of an intro to set theory course, we were introduced the Axiom of Foundation and the Principle of $\in$-induction as one of its consequences. I found it easy to prove that, assuming the ...
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### Axiomatic definition of the complex numbers

The real numbers may be defined axiomatically as a complete ordered field. This description characterises them up to isomorphism. Question: is there a similar way to define the field of complex ...
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### Boolean-valued model, $\mathsf{ZFA}$ and $\mathsf{ZF^-}$

Boolean-valued models are usually used for forcing. We fix a complete Boolean algebra $B$ and inductively define $V^B_0=\emptyset$, $V^B_{\alpha+1}=$ all partial functions from $V^B_\alpha$ to $B$, ...
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### How to introduce non-logical axioms?

Suppose I have a deductive system based on the language $L=\{Variables,\vee,\wedge,\rightarrow,\leftrightarrow,\neg\}$ consisting of propositional tautologies and rules of inference. How do I ...
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### What is the exact consistency strength of this logic of succession and recursion?

The logic of Succession and Recursion is extending first order logic with equality by the following: Add primitives of $N,0,S$ standing for "Natural, Zero, Successor". Also we add primitive ...
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### What are some good references to help explain the need for an axiom schema of replacement rather than an axiom of replacement?

I’m looking for reading material about the philosophical and mathematical issues that may result from using an axiom schema of replacement rather than an axiom of replacement. Among other things, it ...
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### Confusion on Scalar addition in vector space

Consider this problem. Let $V$ be the set of all vectors which are positive rational numbers on which addition of vectors $v,w$ is defined as: $$v+w=vw$$ and the scalar multiplication is the usual ...
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### Is this the Axiom of Infinity?

While studying elementary Set Theory, I came across the Axiom of Infinity. This comes before the book introduces ZFC, so I'm not convinced that it is necessarily the same as the typical definition. My ...
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### How many axioms are in ZFC, and is ZFC a decidable language?

I am reading Jech's book Set Theory and he lists 9 "axioms": Extensionality Pairing Schema of Separation Union Power Set Infinity Schema of Replacement Regularity Choice Two of these are ...
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### Is there any reason why the continuum hypothesis cannot be taken as a postulate which may or may not be true?

I was reading through the Wikipedia article on the Continuum Hypothesis and I was unclear why it couldn't be handled like the Parallel Postulate of Euclid? Is there any reason to believe that there ...
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### clarification on axiom of regularity

I am having difficulty understanding the axiom of regularity (or foundation axiom). From what I read, the axiom of regularity ensures that given any non-empty set $x$, we won't have $x \in x$. In ...
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### $AC_\omega$ is weaker than $AC$, though not provable in ZF without $AC$?

According to the wikipedia article on the Axiom of countable choice, The axiom of countable choice ($AC_ω$) is strictly weaker than the axiom of dependent choice (DC), (Jech 1973) which in turn is ...
Let $\mathcal{S}$ be a complete set of axioms for classical propositional logic (for example, say the Hilbert axioms), and let $\pi\in\mathcal{S}$ be an axiom in it. Is it decidable (in a Turing ...
SUBSTITUTION: Let $\mathbf{F}$ be any algebraic structure. For each $a$, $b\in \mathbf{F}$ if $a=b$ then $a$ can be replaced with $b$ in any mathematical statement involving $a$ and the statement will ...