The list of all whole numbers is infinite. Can numbers start from something trivial such as 0, 1 and then reach infinity?

As per my understanding, infinity is fully formed, perfectly symmetrical, which means to say that a part(small or big) of infinity is also an infinity(like a fractal, wherein each part is the same as a whole). So that would imply that infinity is one as a concept and there are no different classes of infinity, and we tend to classify infinities for the sake of simplification in mathematics.

  • As I understand it, infinity is in mathematics, physics and philosophy a catchy term for pretty much everything we cannot conceive because it transcends our ability to grasp things. Even in formal mathematics, there is positive and negative infinity, see diverging series. – Philip Klöcking Dec 14 '15 at 9:02
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    As you correctly say, this is not the point of view of mathematics.; thus, in mathematics, it is not true that "a part(small or big) of infinity is also an infinity". The set {1} is a part (a subset) of the infinite set N of all natural numbers, but it is not infinite. – Mauro ALLEGRANZA Dec 14 '15 at 9:52
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    Are you referring to the mathematical concept of infinity or infinity in philosophy/logic? If the former, the question would probably be better answered at Mathematics-SE. – fileunderwater Dec 14 '15 at 12:06
  • Numbers don't "reach infinity." Infinity is not a really high number. There are provably different infinities (see Cantor's diagonalization argument). There is also the colloquial concept "infinitesimal," which itself is not an actual mathematical concept. Instead, one talks about values being sufficiently small (see epsilon-delta proofs). – James Kingsbery Dec 14 '15 at 19:49
  • This appears to be a question where the OP could benefit greatly by reading up on infinity in math. (This applies regardless of whether the question is about the mathematical concept or trying to be about an esoteric thing called infinity) – virmaior Dec 17 '15 at 1:39

It would be correct to say: "An infinite set has proper subsets which are infinite again."

Otherwise trivial counter examples exist as Mauro points out.

The concept of infinity as detected by Cantor is much more subtle than a simple argumentation with symmetry captures.

I agree with Mauro that Cantor detected an infinite number of different kinds of infinity. It is a fascinating domain from mathematical set theory. See my comments to @Iowa for some example definitions and results.

  • Please correct me if i am wrong anywhere. As per my analysis, from the point of view of mathematics, the more appropriate term to use would be "Uncountable", which would instantly imply it is finite but very large and starts from a finite point.. That would make perfect sense. Same goes to the size of the universe, which physicists should say, might be very big(uncountable), rather than, might be infinite, which would otherwise imply that every point in the universe would be infinite, and as bright as a surface of a star. – Iowa Dec 14 '15 at 10:56
  • Same goes for Cantor's definition, It should be different kinds of uncountable, rather than, different kinds of infinities. The very definition of infinity is something that is fractal-y infinite or else it can never be infinite. – Iowa Dec 14 '15 at 10:56
  • @Iowa According to set theory: 1) Two sets have the same cardinality if and only if a bijective map exists from one to the other. 2) A given set is infinite if and only if it contains a proper subset with the same cardinality (Dedekind). 3) Equivalently: A given set is infinite if and only if an injective map exists from the set of natural numbers to the given set. 4) A given set is infinite countable if and only if it has the same cardinalitc as the set of natural numbers. – Jo Wehler Dec 14 '15 at 11:17
  • @Iowa 5) Cantor detected that infinite sets exist which are uncountable, e.g. the set of real numbers. 6) Cantor detected that the cardinality of the power set is always strictly bigger than the cardinality of the original set. Hence starting with the power set of the set of natural numbers you get step by step an infinite number of uncountable infinite cardinalities. But note: The class of all cardinals is not countable. – Jo Wehler Dec 14 '15 at 11:19
  • @Iowa "uncountable" has a very specific meaning when you are talking about mathematical concepts of infinity. That meaning is not consistent with your usage. You're free to make up any terminology you please, but its easier to communicate your ideas if they use the typically accepted terminology. – Cort Ammon Dec 17 '15 at 2:33

The list of whole numbers is infinite; can one start from zero, one, two and reach infinity?

Aristotle had a concept of potential infinity, in that one can keep going towards infinity, but never reach it; this concept is the one still used in physical thinking - which is why whenever they find an infinity in a theory it's a sign of something having gone wrong somewhere; one might say potential infinity is then a concept of onto-logic.

Mathematics recently has gone in a different direction, essentially as Wehler points out from Cantors conceptualisation of infinity; but to start this off, they cannot simply go from 0, 1, 2 ...; they need to assert a new axiom:

I: there is an infinite set

So, in a way your intuition was correct.

Infinity, is fully formed and perfectly symmetrical

This sounds rather like Parmenides in-finite; ie not finite; where finitude is identified with the phenomenal world and the non-finite, the Parmenidean One is perfectly 'well-rounded'; ie symmetrical.

But then you say this:

which means a part of infinity is like infinity

Which is how, modernly, we understand we have an infinite set - which was one of the 'inputs' into Cantors theory; I mean it was thinking over and through this paradox that helped form the modern theory of the mathematical infinite.

  • Can the down-voter or some-one sympathetic to the down-vote please explain where my stab at an explanation has gone wrong? Or are they simply prejudicial against concepts of infinity that are philosophical, rather than the merely mathematical mainstream? – Mozibur Ullah Dec 14 '15 at 13:13
  • I am just a sympathetic someone, but to me phrases like "Infinity, is fully formed and perfectly symmetrical" don't make sense.. trying to attach adjectives like "symmetrical" to something that has not been defined in the first place. It may be that you dislike the "mathematical mainstream" approach, but it would be good to provide an alternative, viable approach. Talking about non defined things in hand-waving sort of way it's a waste of time from my point of view. But of course I have a mathematical approach so I'd be happy to change my point of view throgh a discussion :-) – Ant Dec 14 '15 at 18:58
  • @ant: it's not that I dislike the mathematical mainstream - I know it as well as any one, if not more; and I do mention it; spheres are perfectly symmetrical, and for anyone with a knowledge of the history of philosophy, mathematics and physics - the parallels are obvious; my dislike is of only the mainstream; there are alternative insights - it's well to remember that Cantorian infinities – Mozibur Ullah Dec 14 '15 at 19:01
  • was a minoritarian position when it was invented; in fact, Cantor was driven to nervous breakdown by the mathematical conservatism of the time; Brouwer, the inventor of intuitionism was warned by his advisor to get a tenured position before expounding on his alternative mathematical philosophy. – Mozibur Ullah Dec 14 '15 at 19:05
  • If you look at Brouwers original work, to some extent it is hand-wavy - by today's standards, very hand-wavy: the two acts of intuitionism; now we have Heyting algebras...in correspondance to Boolean algebras! – Mozibur Ullah Dec 14 '15 at 19:10

Regarding your first question : "Can numbers start from something trivial such as 0, 1 and then reach infinity?"

This is a mathematical question and the mathematical answer is no. To understand why the answer is no requires some philosophical considerations.

To best way to understand why is to consider Cantor's original "three principles of ordinal generation". (Here, an ordinal is just a certain type of a set.)

The three principles exploit the notion of successor, limit, and supremum. Rather than get bogged down in technical details I will appeal to your intuition here. When we apply any one of these principles to a finite collection of ordinals (numbers) - e.g., {0} or {0,1} - we generate one new ordinal (number). Adding one to a finite number yields a finite number and therefore the three principles of ordinal generation do not allow us to escape the finite domain. At this stage, we cannot take the supremum of all finite numbers to obtain an infinite set since we have not yet generated all finite numbers.

The only way to formally obtain an infinite set is to introduce an axiom that asserts the existence of an infinite set - the so-called Axiom of Infinity.

Once we have asserted the existence of an infinite set, we can continue to apply the three principles to generate new infinite sets which are mathematically distinct from that set identified by the axiom of infinity.

You then say "As per my understanding, infinity is fully formed, perfectly symmetrical". This is not entirely correct. Certainly those sets formed by the three principles are fully formed in the sense of being well-defined. However, in the case of infinite sets, these sets are not symmetrical since, for example, there is a smallest whole number but no greatest whole number. Other infinite sets may have a least member and a greatest member, but "in between" there are subsets with no greatest member.

It is also important to understand that our intuition of the infinite extends beyond those well-formed sets generated by the three principles to what Cantor called the absolute infinite. These collections are not well-formed. These are inconsistent infinities such as the collection of all well-formed sets. The inconsistency of this collection is witnessed by Russell's famous paradox.

Finally, you say "So that would imply that infinity is one as a concept and there are no different classes of infinity, and we tend to classify infinities for the sake of simplification in mathematics." This is not correct as a statement about mathematical infinity. As we have already noted, once we have asserted the existence of an infinite set we can continue to apply the three principles of ordinal generation to generate new sets which are mathematically distinct.

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