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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 a

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

rcfox 9y ago
> 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.
- posterboy 9y ago
  but those are not the intended type of object, because MAX_FLOAT plus n and 1.0/0.0 are ambiguous, unintended values.
  - Retr0spectrum 9y ago
    Why would that matter?
- johnhenry 9y ago
  In context, these are not part of IEEE 754.
poizan42 9y ago
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.
- johnhenry 9y ago
  How do we know that the infinity is countable from the context?
  - poizan42 9y ago
    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
mFixman 9y ago
You can define the sequence in the set ℕ∪{∞}.
deleted 9y ago
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rcfox 9y ago

> 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.

posterboy 9y ago
but those are not the intended type of object, because MAX_FLOAT plus n and 1.0/0.0 are ambiguous, unintended values.
- Retr0spectrum 9y ago
  Why would that matter?
johnhenry 9y ago
In context, these are not part of IEEE 754.

poizan42 9y ago

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.

johnhenry 9y ago
How do we know that the infinity is countable from the context?
- poizan42 9y ago
  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

mFixman 9y ago

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

deleted 9y ago

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