Talk:Supernatural number

problematic article

I think this article confounds two rather different notions. On the one hand, there's the formal product of infinitely many primes; on the other, there's Hofstatder's terminology for nonstandard elements of models of arithmetic, which also seems to be the justification for the claim that supernatural numbers are related to the Goedel theorems.

These are not the same thing at all. I don't immediately have a proposal for what should be done with the article, but it can't stay like this. --Trovatore (talk) 17:08, 26 March 2009 (UTC)

What in the article relates to the latter notion? All I can see is the line about Goedel and the Hofstadter reference (which is mentioned nowhere in the text). The obvious thing to do seems to be to just remove those bits and (if the terminology is notable enough to be mentioned at all; does anyone besides Hofstadter use it?) add a note like 'supernatural number can also refer to the different notion of a nonstandard element in a nonstandard model of arithmetic.' (which article needs work, but that's another matter). Algebraist 17:28, 26 March 2009 (UTC)

The second sentence asserts that [s]upernatural numbers are closely tied to Gödel's incompleteness theorems, though without saying in what way. This is not an incredibly perspicacious statement even if interpreted as referring to elements of nonstandard models, but at least it would make some sense. I'm completely at a loss to figure out what formal products of infinitely many primes have to do with the incompleteness theorems. --Trovatore (talk) 18:50, 26 March 2009 (UTC)

Oh, I guess you mentioned that line. Well, that's true, we could snip that out, but there's not a lot left. Is this a real topic? The only ref is to PlanetMath (which as a tertiary source should not really be a reference); the latter mentions a couple of works in the "Bibliography" but doesn't really say what they say. --Trovatore (talk) 18:55, 26 March 2009 (UTC)

Google Books and Scholar have plenty of hits for the formal product usage. I'm sure that stuff can easily be sourced. Algebraist 19:01, 26 March 2009 (UTC)

Just a comment. $n_{p}$ is either a natural number or infinity. Umm, why can't $n_{p}$ be a supernatural number itself? (Igny (talk) 17:43, 26 March 2009 (UTC))

There's no obvious way to add two of these gadgets, so if you liberalized the exponents in that way, it's not clear how you'd even multiply the remaining objects. What structure is left? Of course you can still discuss them formally, as some sort of generalized graphs or something, but there doesn't seem to be any motivation then to mention the prime numbers or to think of the things as generalizations of the naturals. --Trovatore (talk) 18:58, 26 March 2009 (UTC)

Does anyone have a copy of Hoftstadter to hand? I can't tell from Google exactly what his usage is, let alone who follows him in it. Algebraist 00:35, 27 March 2009 (UTC)

Hofstadter's discussion of "supernatural numbers" comes on pp. 451–455 of the 20th Anniversary Edition of GEB. My very uneducated guess is that the entities he's talking about are what Wikipedia calls hypernatural numbers, i.e. hyperintegers. One of GEB's most interesting and perhaps-helpful-for-identification claims on the subject is:

There is one curious and unexpected fact about supernaturals which I would like to tell you, without proof. (I don't know the proof either.) This fact is reminiscent of the Heisenberg uncertainty principle in quantum mechanics. It turns out that you can "index" the supernaturals in a simple and natural way by assocating with each supernatural number a trio of ordinary integers (including negative ones). Thus, our original supernatural number, I, might have the index set (9,-8,3), and its successor, I+1, might have the index set (9,-8,4). Now there is no unique way to index the supernaturals; different methods offer different advantages and disadvantages. Under some indexing schemes, it is very easy to calculate the index triplet for the sum of two supernaturals, given the indices of the two numbers to be added. Under other indexing schemes, it is very easy to calculate the index triplet for the product of two supernaturals, given the indices of the two numbers to be multiplied. But under no indexing scheme is it possible to calculate both. More precisely, if the sum's index can be calculated by a recursive function, then the product's index will not be a recursive function; and conversely, if the product's index is a recursive function, then the sum's index will not be. Therefore, supernatural schoolchildren who learn their supernatural plus-tables will have to be excused if they do not know their supernatural times-tables — and vice versa! You cannot know both at the same time.

— Douglas Hofstadter; Gödel, Escher, Bach; Chapter XIV

Does this help? --Quuxplusone (talk) 06:29, 12 April 2017 (UTC)

Two more problems

One is easy to solve: we should remove "gcd or" from the fragment "we can now take the gcd or lcm of infinitely many natural numbers to get a supernatural number". gcd of infinitely many natural numbers is still a natural number (easily proved).

The other one is a bit harder: there are, at a first glance, two incompatible notions of "natural numbers" in the article. If we want to represent 1 as a supernatural number, we must allow n_p=0, so in "each n_p is either a natural number", the notion of "a natural number" must include 0. However, then it is not very clear how we can represent 0 as a supernatural number, so that "are a generalization of the natural numbers" part can be true.

It can be done, if only multiplication and lcm/gcd stuff (not order, for example) have to be preserved: represent 0 with all n_p equal to infinity. But this is IMO counterintuitive enough to be mentioned as a separate case, together with the note that we consider 0 to be a natural number here.

I'm not very experienced with Wikipedia style, so I'd like someone else to do the edits, but if they aren't done (and there are no comments) in a few weeks, I'll do them. Thanks. 31.147.115.145 (talk) 03:15, 3 November 2011 (UTC)

There's no reason we couldn't use {0, 1, ...} in one case and {1, 2, ...} in the other. As it happens, there's no need: $1=2^{0}3^{0}5^{0}\cdots$ and $0=2^{\infty }3^{\infty }5^{\infty }\cdots .$ So supernatural numbers do generalize natural numbers, and so there's no problem saying that the gcd in infinitely many natural numbers is supernatural (since it will be natural and hence supernatural). CRGreathouse (t | c) 17:24, 10 January 2012 (UTC)

There's no reason we couldn't use {0, 1, 2, ...} in one case and {1, 2, ...} in the other, but there's a very good reason why we shouldn't refer to them both as the "natural numbers" in one article without clarification, which what the article appeared to be doing at the time the above message was posted. It is true, as the anonymous editor said, that this apparent inconsistency can be cleared up by representing 0 as a product with all indices infinite, but that was not specified in the article at the time, so the point was a valid one. The point about gcd, however, was a mistake. The given construction does indeed produce a supernatural number as the gcd. That supernatural number can naturally be identified with a natural number because all but finitely many indices are zero and none is infinite, but that does not alter the fact that the definition provided defines it as a supernatural number. JamesBWatson (talk) 14:12, 10 February 2012 (UTC)