Things that we don't know exist but they exist, could we say that they exist? We can't think of a big big number, but probably that big big number exists, in the way of atoms that form a giant planet etc...

So, shall we say that numbers are infinite considering my first question and the example I gave?

  • It is hard to say that a number exists "in the way of atoms" do... but - as you say - you can think "of a big big number"; then add one to this big big number : this is the "evidence" for the infinity of numbers, i.e. the possibility of an unlimited repetition of the operation of adding one. – Mauro ALLEGRANZA Sep 18 '15 at 6:43
  • Numbers themselves are not equations. 1 divided by 0 = infinity and is a equation. – Kris Sep 18 '15 at 11:45
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    @Kris, no 1 / 0 is undefined, not infinity. – James Kingsbery Sep 18 '15 at 14:53
  • I cannot understand what is being asked here. The natural numbers obviously include numbers which are so large that no conceivable notation would suffice to name them. – Nick Sep 18 '15 at 20:24

You are not the only one to question the infinite myriad of numbers. In fact, there are entire schools of thought exploring the infinite spectrum of numbers, entire schools of thought exploring the transfinite numbers beyond the infinite spectrum, and entire schools of thought exploring how to do math where infinities do not exist (known as finitist schools of thought)!

Fundamental to the discussion of infinite numbers is the concept of Peano arithmetic. Giuseppe Peano developed a set of axioms for the so called "natural numbers," which are informally defined to be the sequence 0, 1, 2, 3, 4... The axioms are:

  • 0 is a natural number (we declare it to exist, it is a constant)
  • For every natural number x, x = x (reflexive: everything "equals" itself)
  • For all natural numbers x and y, if x = y then y = x (symmetric property of equality)
  • For all natural numbers x, y, z, if x = y and y = z then x = z (transitive property of equality)
  • For all a and b, if b is a natural number and a = b then a is a natural number (equality is "closed")

We then need to define a function S, known as the successor function, so that we can have numbers greater than 0. Informally, S(0)=1, S(1) = 2 and so on.

  • For every natural number n, S(n) is also a natural number
  • For all natural numbers m, and n, m = n if and only if S(m) = S(n) (S is an injection)
  • For every natural number n, S(n) = 0 is false (the successor of a number is never 0... aka 0 is the "first" natural number)

Now we need the axiom that makes your question so exquisitely interesting, the axiom of induction:

  • if f is a function such that f(0) is true and, for every natural number n, if f(n) is true then f(S(n)) is true then f(n) is true for all natural numbers.

That last axiom is the one that causes so much interesting behavior to occur. It's the one that tries to reach towards infinity, and claims to offer ways to grasp it. And, like all axioms, it does not necesarially state that it is "correct," merely that it is declared to be true within the confines of the rules of arithmetic (as defined by Peano).

Much of arithmetic was formalized onto what is known as "set theory," which is the foundation of a great deal of our mathematics because it appears to be fundamental as to how the universe is organized. Sets deal with particular collections of stuff, like "the set of natural numbers that are smaller than 5," which is written as {0, 1, 2, 3, 4}. Peano arithmetic is most commonly mapped onto set theory using the following construction:

  • The empty set {} is declared to be the constant 0 in Peano's axioms
  • The successor function S(n) is defined to be `S(n) = {{}, {n}} (The successor for any number is defined to be the union of the empty set and a set containing the previous number)

That definition sounds a bit obtuse, but it was chosen because it is easy map all of the other Peano axioms onto these two definitions. With this, we gain the ability to use set theory axioms to manipulate "numbers" in very powerful and fundamental ways. One of the most important of these is the concept of the cardinality of a set. This is the "number" of things in a set. Informally {1, 2, 3}, {3, 4, 5}, and {apple, orange, orangutan} all have a cardinality of 3 because they have 3 elements, but {2, 4, 6, 8} has a cardinality of 4.

This is where it gets tricky, because it turns out "the set of all natural numbers" is a valid set, typically represented with a capital N, so we can ask "what is the cardinality of the set of all natural numbers?" The answer is "infinity," and that statement is made as a definition. We define the cardinality of N to be a particular number, known as ℵ₀ which is given the English name "countable infinity." Yes, to mathematicians, infinity is countable, because you can theoretically start at 0, count upwards 1, 2, 3, 4, 5... and "reach" ℵ₀ according to the axiom of induction. There are also uncountable infinities, such as ℵ₁, known as the cardinality of the continuum or the number of real numbers (assuming the continuum hypothesis is true... there's even different opinions on this). There's even a school of thought on "transfinite" numbers which can handle phrases like "I double dog dare you infinity plus one times!"

Welcome to the rabbit hole of infinity in mathematics. We've defined the word to mean something here. It is defined with respect to a set of axioms. Do those axioms hold in "real life?" Most mathematicians find it convenient to presume they do. The computer you are reading this on today was developed using many models from calculus, and calculus's roots are found deep in infinity (particular its concept of "limits). So far, that assumption has done us pretty good. Is that assumption "true?" That's a more complicated question. There are finitist schools of thought which start from the assumption that the number of natural numbers is finite, usually related to the finite capacity of the human mind or the universe in one way or another. If time is finite, and computation is finite, then one cannot theoretically computer "infinity," so they argue it doesn't exist. Are they right? Well, yes... by their definitions, just as the opposing claim is true by the definitions of the Peano axioms and set theory. Both can arguably be true because they each define the word "infinity" to means something ever so slightly different.

As a closing, it may be worth dabbling in linguistic choice: "So, shall we say that numbers are infinite?" We can say a great number of things. Whether those things meet the ideal of truthhood (itself a very hard word to describe formally) depends greatly on one's individual meanings for words. If you accept the definition for "infinity" given by mainstream mathematics, then "numbers are infinite" is true, literally because mainstream mathematics defines "infinity" as such. If you accept the definition given by the finitists, then "numbers are infinite" is false, literally beacuse the finitists define "infinity" as such. You may choose your own definition. It may even be contextual (it is not uncommon to find Christian mathematicians who define "infinity" within their religion slightly differently than they define it within the mathematics, with no ill effects besides two very similar concepts being assigned the same word in their vocabulary).

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  • "there are entire schools of thought exploring the infinite spectrum of numbers". No one can explore the infinite amount of numbers because they are infinite. You would need infinite amount of years and infinite amount of scholars. – John Am Sep 18 '15 at 21:14
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    This answer contains what I assume is an innocent error. The value of the cardinality of the continuum is one of the great unknowns of set theory. ZFC is not strong enough to answer establish a value. To say that "c" is equal to aleph-1 is to assume the truth of the continuum hypothesis. – Nick Sep 18 '15 at 22:34
  • I truly like this answer. As much as anything is what we say it is when there is popular agreement, this answer goes even further to very quickly and clearly give the mathematical framework whereby we both define terms and specifically how infinity is defined using same. +1 – xtian Sep 18 '15 at 23:30
  • @NickR Thanks for the catch! An edit has been put in place! – Cort Ammon Sep 19 '15 at 0:32
  • @JohnAm You can explore them in finite time, as long as you average an infintessimal amount of time on each number ;-) It does raise the question of how thoroughly we explore some of the larger numbers, doesn't it! – Cort Ammon Sep 19 '15 at 0:34

It's generally accepted that the natural numbers satisfy the Dedekind-Peano Axioms (usually just named after Peano because Dedekind gets stiffed). These axioms imply that there are infinitely many natural numbers. And it's not hard to see why: there can't be a largest natural number n, since n+1 is a larger natural number.

More generally, in the standard (ZFC) axioms for set theory we can prove the existence of quite a few infinite sets. This is a bit less helpful for your purposes, since the existence of an infinite set is built in to ZFC as an axiom, but since ZFC is widely accepted by mathematicians and philosophers it's worth pointing out.

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Things that we don't know exist, but they exist. Can we say that they exist?

Well there's a peculiar opening question! Firstly, you have already said that they exist, so yes, you can obviously say that they exist. As to whether that statement has any truth value is different question, the answer to which is... maybe? we don't know. You answered this yourself in your opening line.

We can think of a big number, and that big number probably exists, in the form of atoms.

We can indeed think of a big number, and there is most probably at least that amount of atoms in the universe. It's generally accepted however that the universes energy (at least that sort which is in the form of matter and atoms etc.) is finite, and so you can certainly think of a number of which there is no such quantity of atoms.

Shall we say that numbers are infinite considering the first question and example?

Well, no we shan't. In the sense of your example numbers are explicitly finite. However, you are referring to the number of things, rather than to the number of numbers. There are obviously an infinite amount of numbers, because you can always add 1 to the number you had.

However, it can be argued that there are not an infinite amount of numbers. The Pythagorean's believed that there was a difference between the concept of number, numbers, and values. 1 is the parent of number and not considered an actual number. 2 is two 1's, and also not a real number. 3 is the first real number, then 4, 5, 6, 7, 8, and finally 9. Giving seven numbers in total. 10 is a greater exaltation of 1, since the binary root of 10 (1+0) is 1; and all numbers after 10 are different versions of the basic seven numbers (3 through 9), and their parents (1 and 2). For example 58 (5+8=13, 1+3=4) is a version of the number 4. 43 (4+3=7 is a version of the number 7).

I find it interesting to note that there are nine numbers in the sense that all numbers can be written with nine symbols, and the square of nine 1's recreates all of these symbols, showing that 1 is, in a way, the creator of number:

111,111,111 x 111,111,111 = 12,345,678,987,654,321

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