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Comment by philipov | Hacker News Reader

What is the name for this sequence notation making use of an infinity sign? I don't know how to read that symbol; is it some kind of operator between finite sequences to generate an infinite sequence?

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As explained in the linked page, the third value in the sequence is infinite. The fourth in the sequence is five.

Nitpick: I think you meant "infinity" not "infinite". Infinite is an adjective, infinity is a noun.

I'm not a mathematician but I tend to avoid using the word infinity and prefer "infinite value" as I used above.

In the context of the sequence in the article I think "infinite value" is confusing. Infinity is a better choice. Using infinity signifies that that number of regular polytopes is not finite. The ∞ in this context just means that there is no bound on the number of regular polytopes and not meant to be taken as an actual number. When you say infinite value it makes me think, "Which infinite value?" There are infinitely many infinite cardinal numbers.

Am mathematician.

As ∞ is not the same type of object as 1, 5, 6 3, etc; the sequence isn't technically well defined. Actually ∞ itself isn't well defined as it could be one of an infinite number of cardinal numbers.

Something along the lines of "1 1 \aleph _{0} 5 6 3 3 3...", with each element being a well-defined cardinal number [1], might be technically closer to being correct, but it is still confusing.

(Also, sorry for the TeX).

1. https://en.wikipedia.org/wiki/Cardinal_number

> As ∞ is not the same type of object as 1, 5, 6 3,

Technically they are, if you consider the set of IEEE 754 floating point numbers.

but those are not the intended type of object, because MAX_FLOAT plus n and 1.0/0.0 are ambiguous, unintended values.

Why would that matter?

In context, these are not part of IEEE 754.

From context all the numbers are cardinal numbers since we are talking about the size of a set (i.e. the set of different regular polytypes in each dimension). Also from context the infinity here must be countable, so indeed it must be the sequence "1, 1, \aleph _{0}, 5, 6, 3, 3, 3...". You don't need to spell everything out explicitly as long as it is clear from the context.

If we want to get really formal, then what we have is a mapping from the natural numbers into equivalence classes of sets under the bijection equivalence relation.

How do we know that the infinity is countable from the context?

Well, the argument is that for any natural number n we can make an n-gon. So we have a one-to-one correspondence between natural numbers and dimension 2 polytypes, so by definition the cardinality of that set is aleph zero

You can define the sequence in the set ℕ∪{∞}.

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