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I need some help understanding here..
Consider the set of all integers. Let us write them down like that:
{...,-3,-2,-1,0,1,2,3,...}
Can we call "0" the middle of this infinite set?
What properties does something infinite need to have a center in higher dimensions than a set of numbers?
pic unrelated
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>>9122573
now it's all the way on the left side
{0,1,-1,2,-2,3,-3,4,-4,...}
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>>9122599
so, there's effectively no method of creating an infinite set in the way I described?
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>>9122573
What do you mean by 'center'?
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>>9122650
Center means equi-distant from either side.
With finite sets - or objects - that's pretty straightforward...
Would you consider zero the center of the number line?
Why(not)?
Can one extend the notion of a center in any meaningful way to infinite sets(or objects)?
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>>9122710
i think you can, but only if you consider Z to be an ordered set thats numerically ordered either ascending or descending
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>>9122725
What do you mean by ordered?
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>>9122774
like (a, b, c) != (b, a, c)
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>>9122710
>Would you consider zero the center of the number line?
>Why(not)?
I wouldn't since |you could consider any integer as the 'center', there's the same cardinality of integers on either 'side' of any integer
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>>9122573
You can call it whatever you want. If you want higher dimensions, then consider something like [math] \mathscr B(S_1) [/math] (The set of bounded real-valued functions on the unit circle). This set is infinite dimensional and it's "center" can be considered to be the function [math] x \mapsto 0 [/math]. You should probably look up vector spaces.
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>>9122813
Thank you, I shall go on a Wikipedia dig.
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>>9122573
In an infinite set, the """middle""" is determined by the sequence from which you draw a pattern in reference to what you're asking. Zero would technically be the """middle""" because it is the origin of the set described. If you imagine that you start from nothing and then gain +1, -2, +4, -6, and so on and so forth, you would eventually draw out the entire line described in your inquiry.
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>>9124480
As in: when you go distance d and -d from the element, they add up to the element you started from?
"middle" or "center" seems to me to be about distance, so we still need it to be ordered, for that to make sense...
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>>9122573
0 feels like the center precisely because it's the neutral element of R considered as a vector space. it has nothing to do with the topology or order properties, it's all about algebra. if you forget that it's an algebraically distinguished element, for example by embedding the line in some affine space, there's no natural choice for a "center" unless some further structure has been defined.
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>>9124532
Yeah pretty much. The only meaningful difference between 0 and any other integer is algebraic, if you get my drift. You can put a norm on the group if you want to, it doesn't change anything but gives you what you want.
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>>9124593
>>9124656
Thanks, that satisfies my curiosity sufficiently. For now.
Have some more triangles.
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>>9124682
Thanks mate
Thread posts: 18
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