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The same idea as other answers, but a different explanation (which makes more sense to me, at least): there is no such thing as true, period. A statement can only be true or false in some model, which consists of a certain set of "original" statements whose truth is assumed, and everything provable from those statements. The "original" statements are the ...


28

You're treating the word "axiom" as you were probably taught in high school. That an axiom is something which is "simply true as an assumption". Modern mathematics has changed that definition to "an assumption made in a certain context". Not every axiom is called an axiom, some axioms are proved as theorems, and sometimes lemmas are used for axioms. And not ...


6

The existence of a model of a statement does not mean that statement is "true" (whatever that means; see below). For example, the Poincare disk is a model of Euclid's first four postulates plus the negation of the parallel postulate; this does not mean that the parallel postulate is "false." What having a model of a set of statements does mean, is: that set ...


1

All I can do is give you some pointers to the standard terminology of elementary model theory and universal algebra: a structure consists of a set equipped with a set of finitary operations and relations (https://en.wikipedia.org/wiki/Structure_(mathematical_logic). An algebraic structure is one in which the only relation is equality (which many authors ...


4

If you accept the following: The standard ordering among real numbers agrees with the standard ordering among natural numbers, i.e. nonnegative integers. This takes some work to prove rigorously, but the sociological proof is simple: Any sane mathematician would refuse to accept as standard any ordering among real numbers that didn't agree with the ...


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To my knowledge, if we want to tell a tale about the construction of the real numbers, it would go like this: Natural numbers are defined using set theory: $0 = \{ \}$, $1 = \{ \{ \} \}$, $2 = \{ \{ \}, \{ \{ \} \} \}$, and so on. "$+$" is a binary operation defined on $\mathbb N$. Through it, we define the negative integers as the inverses with respect ...


14

You ask that as a foundational question. The answer is simple. It depends on your foundational approach to the real numbers. We can begin by constructing the natural numbers, then the integers, then the rationals, and then the real numbers by one mechanism or another. In this approach, the order on $\Bbb N$ is extended in a way which is compatible with the ...


4

Usually, the field of real numbers is constructed by taking Cauchy sequences of rational numbers. The rational numbers have an ordering derived from the ordering on the integers: $p / q > r / s$ just in case $(p / q) - (r / s)$ has either a positive numerator and denominator, or a negative numerator and denominator. You can then get the ordering on the ...


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The ordering on the reals comes in the end from that on the integers, via that on the rationals. The ordering on the integers is the unique transitive relation satisfying $0<1$ and $a>0\implies (b>c \iff b+a>c+a)$ and not permitting $a<b$ while $b<a$. Then the ordering on the rationals is determined, though I think you need to separately ...


1

The axiom of pairing is fine. That's not the issue. You're confusing between the Russell paradox and a construction of sets which "look like" the class defined in Russell's paradox. Note the subtle difference here. The classical Russell paradox is that $\{R\mid R\notin R\}$ is not a set. But here we only use restricted comprehension (also known as ...



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