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Axioms are not "defined to be true"; I'm not even sure what that would mean. What they are is evaluated as true. Practically speaking all this means is that in the mathematical context at hand, you're allowed to jot them down at any time as the next line in your proof. Definitions have virtually nothing to do with truth, but are instead shorthand for ...

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A definition is a conservative extension of the language by a new symbol and some axioms involving this symbol. The key word here is conservative; in general axioms strengthen the system in question, while definitions are not allowed to do so.

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In this instance, I'm taking Peano arithmetic to be defined in the first-order theory over functions $0, s, +, \times$ of arity 0, 1, 2, 2 respectively. The symbol $0$ is just that - a symbol. It needs no definition in this language. It already exists. We need the axioms to tell us what we're allowed to do with these symbols. You want to "define" how $0$ ...

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From a proof theory perspective, there is no difference. They are both effectively announce the truthhood of something without providing a proof thereof. The difference arises when one applies model theory, which is required to apply the mathematical results (whether applied explicitly or implicitly). A definition is wholly contained within the ...

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Saying something cannot be proven by any other means but induction is quite a bold statement, and I am not capable of giving it. It is also not an entirely well defined statement, unless you formally define all your axioms (one of which is induction) and then say "which statements cannot be proven if I remove induction from my set of axioms?" But I ...

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A simple approach is inspired by : George Tourlakis, Lectures in Logic and Set Theory : Volume 2, Set Theory (2003), page 130. The formula $\exists x (x=x)$ can be derived from axiom : $\forall x \varphi(x) \to \varphi[x/t]$ with $\varphi(x) := \lnot x = x$ and $x$ as $t$, by tautological implication [$\vDash_{TAUT} (\mathcal A \to \lnot \mathcal ... 4 Question 2 and 4: this is legit. If you prefer to be concrete, first invoke the axiom of infinity (as you say), which states that "there is a successor set", and then use$A_0$as shorthand for that successor set. (In this proof, it's just assumed that some set exists. To be ultra-formal, you're right: it has to be proved.) You could look at it like this: ... 3 See : Stephen Cole Kleene, Mathematical logic (1967 - Dover reprint), page 289 :$\cfrac{A, \Gamma \to \Delta, B \quad \quad B, \Gamma \to \Delta, A}{\Gamma \to \Delta, A \equiv B} \equiv \text{: right} \cfrac{A,B, \Gamma \to \Delta \quad \quad \Gamma \to \Delta, A, B}{A \equiv B, \Gamma \to \Delta} \equiv \text{: left} $If ... 3 I found exactly the same question on Quora and I am just copying the answer given by David Joyce, Professor of Mathematics at Clark University. I hope you will find it useful. Here's the link to the original answer. Axioms come mainly in two different kinds—existential and universal. They often go along with definitions. For instance, an existential ... 3 Consider a similar example from set theory : Kenneth Kunen, The Foundations of Mathematics (2009), page 10 : Axiom 0. Set Existence :$\exists x(x = x)$. Axiom 1. Extensionality :$\forall z(z \in x \leftrightarrow z \in y) \to x=y$. Axiom 3. Comprehension Scheme : For each formula,$\varphi$, without$y$free,$\exists y\forall x(x \in y ...

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Definitions aren't used to say things exist or something is true about things. They're used to make it easier to talk about things. where as an axiom is a statement or proposition which is regarded as being established, accepted, or self-evidently true.

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Here is my simple explanation of what I think the difference is between a definition and an axiom: An axiom is a rule or a "law of the land" that we decide we will follow/enforce. For example, the axiom of choice in set theory says if you have any arbitrary collection of non-empty sets, you can always form a new set by picking one element from each of the ...

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A definition is a choice to call something by a specific name/reference/identifier/pointer. What is a name? A cognitive synonym for what people understand to be "equivalent". That is a long philosophical discussion. In any sense and reference, Sinn und Bedeutung for Frege, what part of a name attaches to the thing being named? None. A name is an ablative ...

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In a formal treatment of a mathematical theory, definitions are well formed formulae. See this post and this one. A definition in the first order language of arithmetic (with : $0, S, +, \times$) introduces a new symbol, like : $1$ (a constant), or $\le$ (a binary predicate). We can define $\le$ with the formula : $x \le y ... 2 Rewrite $$(x^2-1)=(x-1)(x+1)$$ If 8 does not divide$(x^2-1)=(x-1)(x+1)$, then 2 does not devide$x^2-1=(x-1)(x+1)$. But that is the same as saying that neither$x-1$nor$x+1$are even. Hence$x$must be even. 1 8 does not divide$x^2-1$is the same as "if 8 divides$x^2-1$, then x is odd" which is the same as: $$8|(x^2-1)\rightarrow 2|(x+1)$$ but $$x^2-1=(x-1)(x+1)$$ Then since$8=4*2$Two divides both$x-1$and$x+1$, and we proven our case 1 When dealing with collections of random variables indexed by time, we need more than just a probability space$(Ω, \mathcal{F},P)$to capture the notion of 'information knowable at time$t$'. It is termed a filtered probability space, which is a 4-tuple$(Ω,\mathcal{F},(\mathcal{F_t})_{t\geq 0},P)$where the new ingredient is a filtration, i.e. a collection ... 1 The set of positive multiples of$11$can be put in bijection with the set of positive numbers, simply by dividing by$11$. If the former is finite, so is the latter. Can you prove, by contradiction, that there are infinitely many positive numbers? 1 You've tried to prove the converse. The contrapositive is instead "if$x$and$y$have opposite parities, then$x+y$is odd". Let$x$and$y$have opposite parities. Then one is odd and one is even (by, I suppose, something like the pigeonhole principle). Therefore by the division algorithm we may write$x = 2k, y = 2m+1$(wlog that$x$is the even one). ... 1 A definition is just a name you attach to something. For example you can attach the name "zero" to something. But for the definition to make sense that "something" must exist and that is usually guaranteed by some axiom or theorem. 1 In a sense one can say that are just different forms of postulation, one defines definitions and one defines axioms (see for example equivalence between natural deduction systems and formal axiomatic systems). In a further investigation, there can be (more or less) differences, between the two. For example: Definitions define new concepts (based on ... 1 In set theory the set$\omega$of natural numbers is constructed on base of the axiom of infinity. This as the smallest set that contains$\varnothing$as element and is closed under the operation$a\mapsto a\cup\{a\}$. After that several properties of natural numbers are proved by induction. For instance the fact that all natural numbers are transitive ... 1 Using base set$\{\alpha,\lnot\alpha\}$, from 1. deduce$\beta\to\alpha$and$\beta\to\lnot\alpha$, then apply 3. with$\phi=\lnot\beta$. 1 If you by "base" set means the left side of the$\vdash$and you get to use the deduction theorem you can simply use the fact that with contradictions in the premise anything can be proved via reductio ad absurdum (ie axiom 3):$\alpha, \neg\alpha, \neg\neg\beta \vdash \alpha\alpha, \neg\alpha \vdash \neg\neg\beta \rightarrow \alpha \$ and similarily ...

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