It is said that our current basis for mathematics are the ZFC-axioms. Question: Where are these axioms in our mathematics? When do we use them? I have now studied math for a year, and have yet to run into a single one of these ZFC axioms. How can this be if they are supposed to be the basis for everything I have done so far?...Read more

When reading about some open problems, a lot of them have quotes by renowned mathematicians that “[the conjecture] cannot be solved using the current technology” or something along these lines. What do they mean by that? Are they talking about the axioms? Or are they generally speaking in terms of intelligence and mathematical abilities?These ones are just at the top of my mind:https://terrytao.wordpress.com/2011/08/25/the-collatz-conjecture-littlewood-offord-theory-and-powers-of-2-and-3/https://m.youtube.com/watch?v=MXJ-zpJeY3E (skip to the en...Read more

Let T be a true theory of arithmetic to which the incompleteness theorems apply. Consider two sentences in the language of T to be equivalent if they are provably equivalent over T. How many equivalence classes are there, under this relation, that contain a true-but-unprovable sentence?...Read more

I'm trying to translate http://www.stat.umn.edu/geyer/nsa/o.pdf to Coq, but I'm stuck right at the start on the simple axioms for the predicate "standard" over natural numbers. The pdf linked here gives these as the first three axioms:$0$ is standardIf $n \in \mathbb{N}$ is standard, then so is $n + 1$.There exists an $n \in \mathbb{N}$ that is not standard.For which I have two potential translations into Coq:Axiom std : nat -> Prop.Axiom std_0 : std 0.Axiom std_Sn : forall n : nat, std n -> std (S n).Axiom exists_non_std_n : exists n : n...Read more

In Tao's analysis volume 1, I am introduced to this thing called the axiom of substitution. While constructing real numbers from rationals, he defined reals to be formal limits of Cauchy sequences of rationals. He said $\lim a_n=\lim b_n$ iff $(a_n)$ and $(b_n)$ are equivalent sequences. Then he defines addition of reals as - $\lim a_n + \lim b_n=\lim(a_n+b_n)$. Then he verifies that the axiom of substitution is not violated i.e. if $x=x'$ then $x+y=x'+y$. (I like to state this as "addition is well-defined"). My question : It seems that the axi...Read more

According to Wikipedia, Godel's incompleteness theorem states: No consistent system of axioms whose theorems can be listed by an "effective procedure" (essentially, a computer program) is capable of proving all facts about the natural numbers.This obviously includes our current system. So has anyone proposed any additional axioms that seem credible?...Read more

I was watching a video where Wolfram was discussing the development of Mathematics, he said something along the lines of:" There is a whole universe of possible mathematics. I was curious about this question for logic for example. We always think of logic as being this absolute thing. But in fact, it is just a particular axiom system that lives in the space of all possible axiom systems..." He goes on to say that depending upon how we enumerate this space, logic is the 50 thousandth posssible axiom system. I was wondering, how would we enumerat...Read more

Note that I am not talking perse, about the definition of axioms. I know what that is, their these 'self-evidently true' statements which are the building blocks of all reasoning, meaning there untraceable. This makes sense, because you can't have an infinite regress where reason after reason, you justify yourself perpetually. However, given the nature of axioms, how do we know when we have one? How do we know that something is 'self-evident' enough (though some may argue it need not be) and that it cannot be proven by any other truths? I think...Read more

Most mathematical structures are defined according to axioms. e.g. we state: Definition. Monoid. A monoid is a tuple $(S,\cdot)$ where $\cdot$ is a binary operation $S\times S\to S$ that satisfies the following axioms: Axiom 1. Associativity. For all $a, b, c \in S$, it holds that $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ Axiom 2. Identity element. There is an element $e\in S$ such that for every $a\in S$, we have $a\cdot e=e\cdot a =a$. But we could also have written this in the following setup, without using "axioms". Definition. Associa...Read more

My question is only indirectly about the axiom of choice, I just happened to come to the question via the axiom of choice and will use it to illustrate my problem.So far I thought axioms were propositions that were a) asserted to be true and b) consistent with the remaining theory.Now I read on the axiom of choice, and something feels strange about it. Isn't there a difference between an axioms such as$$0 \textit{ is a number}$$$$\textit{if n is a number then s(n) is a number}$$on the one hand axioms such as$$\textit{for any relation } R, \text...Read more

As we know a statement may not be proved in some axiom system according to the godel incompleteness theory, can we always solve it by some way that change the axiom system?...Read more

Here, I (basically) stated the group axioms as follows.$(xy)z=x(yz)$$xe=x, ex=x$$xx^{-1}=e$In that post, answerers Martin and Ittay were critical of the above list for not including $x^{-1}x=e$, even though it follows from the above three. Pece's answer also included $x^{-1}x=e$ without comment.How can I tell whether a system of axioms of 'complete'? Is this even a rigorous concept?...Read more

I've been trying to study the field axioms in order to eventually go through Spivak's Calculus (I'm not a math major, but just interested). I noticed that different books have different axioms. For example, Spivak lists the order properties as: $a = 0$, $a$ is in $P$ (positive numbers), or $-a$ is in $P$$a$ and $b$ are in $P$ $\implies$ $a+b$ is in $P$$a$ and $b$ are in $P$ $\implies$ $ab$ is in $P$He then defines $a>b$ as meaning $a-b$ is in $P$. However, another book I looked over stated the order axioms as:either $a = b$, $a<b$, or $b&...Read more

I'm trying to get an overarching understanding of the components of mathematical systems so that in my self study of each category of math I can break them down by their unique aspects, i.e. the operators they use, the major concepts they deal with (i.e. how calculus is about "change"), etc.As far as my experience with formal math terminology goes, im rather weak, and i get utterly confused by the technicality required in formal definitions.As a good starting point, I'd like to better understand what the difference is between an axiom, a theore...Read more

We can form axioms of Boolean algebra or Set theory by forming some expression like:$a=f(a,b,c,d)$Which are sort of replacement rules on expression trees. Now people say arithmetic is built on the axioms set theory. But it must also be built on the language that expresses the axioms in the first place.But this language can express many axioms. Thus it must be able to express axioms that are not part of mathematics.A program like Coq or Isabelle Proof Assistant is built on a language that allows one to express axioms. Why isn't this language its...Read more