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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In bra-ket notation, what axiom allows "distributing" the conjugate ($^*$) across a sum of bras or sum of kets?
An exercise says,
Using the axioms for inner products, prove
$$\{\langle A| + \langle B|\}|C\rangle = \langle A|C\rangle+\langle B|C\rangle.$$
The two axioms I've been given are
They are linear:
$$...
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Power sets vs definable power sets in the minimal standard model of ZFC
As noted in this answer by Eric Wofsey, as well as the Wikipedia page on the Constructible Universe, we have that $L_\alpha$ is strictly smaller than $V_\alpha$ for any $\alpha > \omega$ unless $\...
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Is the "commutativity of addition" axiom in a field redundant?
I have seen an assertion that the "commutativity of addition" axiom in a field is redundant because it can be proved from the other axioms. There are relevant posts here and here.
However, ...
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Real numbers cannot be constructed?
Sorry if this is another crank finitism question, I am bit confused.
According to wikipedia:
Constructivism asserts that it is necessary to find (or "construct") a specific example of a ...
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Confusion of the Von Neumann universe universe
edit: I think Mendelson’s Introduction to Mathematical Logic , section 4.6.5 “set theory with urelements” solves my confusion.
I am very new to the NBG set theory, so this may be a very naive ...
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Set theory: Urelements and classes, where to planets and moons fit?
As I understand it, urelements can be members of sets but sets can't be members of urelements. I'm not sure how to describe this situation. How does ZF fit in a scenario I'm trying to explain (open ...
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Do we really need the axiom schema of replacement?
I'm repeating a bit of set theory these days and was a bit perplexed by the axiom schema of replacement, for which I can't really seem to find a substantial application. After all, if $F: A \to B$ is ...
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Using field axioms to prove the next
how can it be proved using field axioms that
$\frac{1}{\sqrt[3]{100}}=\frac{\sqrt[3]{10}}{10}$
I have the next sketch proof:
First I applied the definition of quotient. Then I used that $1=(\sqrt[3]{...
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Theorems (or references to analysis) of a particular Hilbert-deductive-system using $\\{\neg, \wedge, \vee, \rightarrow\\}$ as primitive symbols?
Context
The System CL
In section 6.3 of Topoi, Robert Goldblatt describes a Hilbert-style deductive calculus (the only inference law is modus ponens) for the ...
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Pasch's axiom, intermediate value theorem and Jordan curve theorem
All of these three conclusions seem "obvious" to laypeople, and they appear to be somehow connected. I am wondering whether Pasch's axiom can be viewed as a corollary of intermediate value ...
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What if "ZFC +CH is false" has no set universe which interprets it as true?
We have proved that "CH is false" is consistent with ZFC. So assuming ZFC is consistent itself, "ZFC+CH is false" is good as an axiomatic system.
But is it also possible that this ...
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Gödel's second incompleteness theorem and Consistency.
According to Gödel's second incompleteness theorem, no consistent axiomatic system which includes Peano arithmetic can prove its own consistency. As I understand it, this result contributed to spark a ...
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Peano axiom of induction with "no junk"
In this Wikipedia treatment of Peano Axioms, if you go down to the first picture you'll see a circle of dominoes and a straight line of dominoes:
The caption says the straight line of dominoes
The ...
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Is dependent choice what one must use in this step of Artin's construction of the algebraic closure?
In both Lang's Algebra and Atiyah, MacDonald, Introduction to Commutative Algebra, the following construction of the algebraic closure is atributed to Artin:
(Atiyah, MacDonald, exercises of Chapter ...
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What is the mathematical definition of "standard arithmetic/standard natural numbers"?
As a consequence of Godel's incompleteness theorem, no axiomatic system can be the definition of standard natural numbers because any axiomatic system of arithmetic will always be satisfied by non-...
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Characterizing "almost-vector spaces" that don't assume the axiom $1\cdot v=v$ for scalar multiplication
Consider a structure satisfying all the axioms of a vector space except for $1\cdot v=v$. I don't know if there might might be a name for these, but lest call them "almost-vector spaces". ...
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Infinite axioms/Group of axioms
When we talk about an axiom, shouldn't it be a group of axioms since we have an axiom for each variable?.
For example, "B ⇒ (C ⇒B)" is an axiom schema standing for an infinite number of ...
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How to show that an unbounded set does not exist using the axioms of ZF Set Theory?
I am currently studying a course on ZF Set Theory and would like to know how to show that a set does not exist using the axioms.
For example, the following two sets are clearly equivalent to a set of ...
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Are there doubtful axioms?
Over my life, I've encountered three different definitions for mathematical axioms:
Axioms are statements that must be accepted on faith. Unbelievers shall be punished by eternal damnation grade ...
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Expressing the instance of a ZF Set Theory axiom for a given property
I am currently in the process of studying ZF Set Theory (without the Axiom of Choice) and I have come across a type of question that is unclear to me. The basic format of the question is to "...
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Why is the Axiom of Regularity preferred over the Axiom (schema) of Induction, or the claim that no downward infinite membership chain exists?
Most of the axioms in $\text{ZFC}$ seem intuitive and sensible to me, including the historically contrioverial Axiom of Choice, yet I struggle to find the Axiom of Regularity as intuitive as the rest. ...
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Set Theory Axiomatics
Thinking about Modal Logic made me wonder about Euclidean relations in general, and how they might show up in other areas of math, like Set Theory. I began to think of some examples of Euclidean ...
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What does it mean to discard a hypotesis in a natural deduction?
My textbook says that in the deductive system of natural deduction every hypotesis must be discarded by a rule of inference and after being discarded, it cannot be used again in the deduction... But ...
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Uniqueness of standardization
So I'm reading Kanovei's and Reeken's book "Nonstandard Analysis, Axiomatically" and there is something which I'm not quite understanding.
So, in page 15, it is stated that a "...
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Is $\forall x (\phi)$ a formula in ZF even if $\phi$ does not contain x?
I was reading A Quick Introduction To Basic Set Theory by Anush Tserunyan, and in definition 1.1, the author defined a notation of a formula in ZF with few criterias. One of them states:
If $\phi$ is ...
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How do we know whether a certain statement is provable or not?
Certain statements are known to be unprovable within a given axiomatic system; the continuum hypothesis within ZFC is an example. We can either add the continuum hypothesis, or its negation, to ZFC, ...
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Why does this description of vector spaces include so many tedious/ seemingly obvious criteria?
I'm a CS student. I commonly notice that when I'm learning math-heavy topics (like machine learning) that the descriptions often seem overly tedious and unintuitive, like the one below.
Vector spaces ...
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Define X=R\{k} and define ⋆ to be the operation such that x⋆y=(x−k)(y−k)+k. Does this satisfy closure?
Define X=R{k} and define ⋆ to be the operation such that x⋆y=(x−k)(y−k)+k. Check each of the four axioms of a group (closure, associative, identity, inverse). Which of them hold? Is (X,⋆) a group.
It ...
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Stronger versions of the Power Set Axiom?
In this article on Skolem's Paradox, it gives the Power Set Axiom as an example where models may badly misinterpret the axioms (compared to the "intention" of the axiom). While originally I ...
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Can we axiomatize the complex numbers without directly defining the reals?
I've decided to attempt the entire Rudin sequence in a single 6 month period, because I'm insane. Rudin spends very little time on foundational matters, and that bothers me, it makes the subject of ...
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Equational axioms for quantifiers
I seek an equational axiomatization of the quantifiers of predicate logic (that permits empty domains).
Start with the equational axioms for Boolean algebra. Add the following axioms, which come in ...
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How do we know inference rules are correct?
I know axioms are statements which are assumed to be true (meaning that axioms are not proved).
Theorems are statements which can be proved or has been proved. In the proofs of theorems we can use ...
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Modifying the axiom of reducibility in Friedman's K(W)
Let RED2 be the axiom schema
$$\forall \bar{x} {\in} U (\phi \to \exists u {\in} U \phi[U := u])$$
where $\phi$ is a formula with free variables among $\bar{x}$.
In Friedman's K(W), how does replacing ...
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Is the class of ordinals part of the class of sets in NBG set theory?
In Von Neumann–Bernays–Gödel set theory (NBG), the fundamental objects are classes, not sets. But, in addition to classes, in NBG we have some sets. I took for granted that in NBG we have the ...
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Axiom of foundation implies that bottomless sets are empty?
In the notes that I am using to study Set Theory, the Axiom of Foundation is presented as follows:
Axiom of Foundation:
$$\forall x ( \exists y (y \in x) \rightarrow \exists y ((y \in x) \land \...
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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 ...
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Decidability of redundancy of axiom in classical propositional logic
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 ...
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Is "substitution" an axiom?
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 ...
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