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We need to assume $\neg a$ is an abbreviation for $a \to \bot$; otherwise, there is no way to introduce negations without having an assumption with a negation in it. First note that $b \to a, a \to \bot, b \vdash \bot$ by two applications of modus ponens. Then we do three applications of the deduction theorem: first we obtain $b \to a, a \to \bot \vdash ... 0 The basic problem with your plan is that your$E$does not necessarily constitute a set in a model of ZF. Sure, it will exist (and then equal$S$) in the intended interpretation of set theory, consisting of some Platonic universe of sets. But if ZF is consistent at all, it will also have "non-standard" models where the elements of the form$0, s(0), s(s(0))$... 3 The class of limit ordinals cannot be a set. Its union would be a set of all ordinals, and then the Burali-Forti paradox obtains a contradiction. The class wouldn't qualify as an ordinal even it if were a set, because an ordinal is supposed to be transitive. But, for example, the class of limit ordinals contains$\omega$but not its element$42$, so it is ... 2 The issue is essential incompleteness. If the theory is incomplete, which means that there are well-formed sentences that are neither proved nor disproved by the axioms and rules of the theory, there could be a complete extension and in that case a definition of truth. For example, the first-order theory of groups is undecidable (hence incomplete), but ... 29 The concept you have defined is usually called "provable", not "true". This is indeed a quite central and important concept, but the point is that it is different from truth. For example, let$\psi$be the formula$(\forall x)(\forall y)(x=y)$. Then$\psi$is not provable in your system, and$\neg\psi$is not provable in your system either (these two facts ... 3 I think what is failing is that your definition above is that you are talking about formulas, which are within a syntactical context while "truth" is a semantic notion (i.e. it depends on the model you are interpreting those formulas) but you can talk about logical validity which holds for example for your logic axioms since they are true in every possible ... 2 Back to the original question: Does Mathematics Require Axioms? The best answer I can think of is: Not at all - until they do. In actual practice, working mathematicians go about developing new mathematics using tools of ordinary human thinking and speaking - they model abstract objects as pictures (in the head or on the board); they 'look at the objects' ... 3 The relevant sense of comprehend is ‘to include, comprise, or encompass’, which is indeed the sense reflected in the adjective comprehensive. The axiom schema of comprehension allows us to form sets that comprise or include all elements of a given set that have some particular property. (It’s also called the axiom schema of specification: we form sets by ... 1 As a simple implication of CH, I can say that in general topology, M. E. Rudin proves that if CH is true then$\mathbb{R}^\omega $with box topology is a normal topological space. [General topology and its applications, vol.2, 1972] 3 In two dimensional space, if we want a vector corresponding to a given angle$t$, we can do:$(x, y) = ( \cos(t), \sin(t) )$But what about the reverse? We have a vector, and want to know the angle. A common solution is to use arctan:$t = \arctan( \frac{y}{x} )$Here$\frac{y}{x}$represents the gradient of the vector. So it has a meaningful value ... 13 Rather than viewing division as an operation in its own right (that would take a dividend and a divisor to a quotient), mathematicians think about inverses of multiplication. So one thinks of$\frac x y$as$x · y^{-1}$where$y^{-1}$is, by definition, a number inverse to$y$, i.e. fulfilling$y·y^{-1} = 1 = y^{-1} y$. For example$\frac 3 2$is rather ... 2 I gave an answer to a similar question some time ago here. To put it briefly, if you are ever solving an equation in one variable by just addition, multiplication, subtraction, and division, and you end up with a contradiction, then you can conclude with certainty that the one time you divided both sides by an expression involving a variable, the expression ... 6 Theory of relativity. $$m = \frac {m_0}{\sqrt{1 – (\frac vc)^2}}$$ This means that as your velocity (speed) increases, and gets closer and closer to the speed of light, your mass increases (therefore, mass is related to velocity). It also proves, that it is impossible to travel faster than the speed of light. If an object were to do that, its mass would ... 1 Mathematicians occupy themselves with questions like 'If fact A is true, what else turns out to be true?'. Once you get started, one fact leads to another and another. However, you have to get started somewhere. Axioms are the things we assume to be true at the start of a thought process so that we can proceed to explore the consequences of those ... 2 In simple terms, an axiom is a statement which is widely recognized to be true. In mathematics this statement is something like the fundamental theorem of algebra, whose validity is is hardly worth questioning. 3 Though not actually correct, I've seen it used to "trick" people with a proof that 1=0 Consider two non-zero numbers x and y such that$x = y$Then$x^2 = xy$Subtract the same thing from both sides:$x^2 - y^2 = xy - y^2$Dividing by$(x-y)$, obtain$x + y = y$Since$x = y$, we see that$2 y = y$Thus$2 = ...

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Axioms are rules, and the game is to show what's allowed by the rules. Just like rules, axioms are true because they we say they are true. Also just like rules, we can impose whatever axioms we want! In the real world, the rules we follow are generally the ones everyone agrees upon (like not stealing) rather than some ones people make up arbitrarily. ...

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Is the basic definition of a derivative not an example of ${\frac a0}$? $$\lim_{h\to 0} {{f(x+h) - f(x)} \over {h}}$$

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This may not be what you would consider an "application', but for computer floating point arithmetic, division by zero is useful for setting up three special values: 1.0/0.0 gives +inf, which is a valid floating point value satisfying the usual extended number line properties. -1.0/0.0 gives -inf, and my all-time favorite is 0.0/0.0, which gives NaN ...

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For consideration, both describe declarations that join primitives in sweet reason. The choice of word depends on attitude. Postulate is a more affixative term, axiom is more transitive. In history, the use of axiom gained favor by mathematicians challenging the permanence of posted speculation (e.g. finite observations about infinity)...thus opening the ...

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My favourite axiom demo is Peano's first which reads Zero is a number. It is clear and obvious to be true but clearly difficult (pointless?) to prove. There's also the Reflexive Property which states: For any quantity a, a = a. Although this may not (technically) be an axiom it has all the properties

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From a mathematical point of view, axioms are the fundamental self-imposed rules of the games which mathematicians play. For example, a considerable part of mathematics deals with structures (think of numbers) for which the rule holds that you can replace $a·b$ with $b·a$. From these fundamental rules, all other rules can be deduced – actually that’s already ...

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Things whose existence is taken for granted either due to common past experience and practice or a given thing handed down to us and unquestioningly adopted. When questioned, one cannot give a previous basis (blinks, scratches head etc.) Often the axiom is at a foundational or fundamental level as a sort of primordial premise for consequential if-then ...

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When you want to build a house, you have to put in a foundation first, no matter whether this house is a mansion or a prefab hut. But what do you put the foundations on? There has to be a building site - some ground that is marked out and fenced off and leveled. This house is our theorem or lemma; we add wing extensions as a corollary. The foundation is our ...

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layman's terms Axioms are things you have to assume to get started thinking about something.

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To a mathematician, or a philosopher, in the pre-modern era (and to some crackpots of the modern era) an axiom refers to a truth so basic that it cannot be proved or argued about from other truths. A basic elementary fact to 'obviously' true that the only way to argue about it is to tautologically proclaim "it simply is true!!!". For instance, the fact that ...

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In mathematics, every result known descends from something else: it is proven to be true from other facts. The one exception is axioms: these things we choose to accept without proving them. We have to choose some axioms, since we cannot prove anything with nothing, but we try and make them as simple and obvious as possible. For example, Euclidean ...

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In former times, axioms were considered to be statements that are so simple and "obviously true" that they cannot be proved (or any attempt to prove them would need to be based on more complicated things - and why bother proving it at all if it is obviously true?). In today's understanding, an axiom is a statement that is, for the sake of developing a ...

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You said you wanted an application. Inspired by the example from Exceptional Floating Point, consider the parallel resistance formula:$$R_{total} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}}$$This formula tells you the effective electrical resistance of a path when the current can choose two routes to take. Let's pretend that $R_1=0$. Then we have:$$... 23 The algebraic structure "wheel" is an algebra with division by zero. The one point compactification of the complex plane into the Riemann sphere almost produces a wheel (one still needs to adjoin the element 0/0). 33 In complex analysis, we talk about the value of a function at infinity. To evaluate f(z) at infinity, compute f(1/z) then plug in 0. This allows us to talk about things like the order of zeros and poles at infinity. Example:$$f(z) = \frac{az+b}{cz+d}$$with ad-bc \neq 0 and say a,c \neq 0.$$f(1/z) = \frac{\frac{a}{z}+b}{\frac{c}{z}+d} = ...

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Yes, in projective geometry or hyperbolic geometry for example, you can see applications or geometric entities that are $\frac{a}{0}$ or just $\infty$ . Generally, non-euclidean spaces have such type of entities or applications.

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