The Corroboree

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So Peano naturally assumes that the sequence of integers goes something like

dude

dude plus one

etc

but can you actually PROVE that there is no integer between 0 and 1?

>>> I Don't think so!

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0.5

next.

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Your posts are awesome lately Thelema.

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0.5 is the whole of something, even if it half of something else, as 1 is also a whole and a half (of 2).

The question is, can there be an infinite zero?

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but can you actually PROVE that there is no integer between 0 and 1?

>>> I Don't think so!

Only by the definition of integer being a number from the set {-oo.... -2, -1, 0, 1, 2.....oo}.

Although - I think that our mathematical understanding is limited by the fact that our counting system is based on integers. We have a discreet number of fingers, there are a discreet number of apples on the tree etc... and our counting system is based on this.

Mathematical constants don't confom to our integer based system, so we either estimate in relation to our integer system (ie: pi can be notated as 3.14..... but will never be completely accurate or use a symbol in place to convey the fact that we can't accurately notate the number using our system.

Perhaps there is a counting system we haven't yet discovered, not based on integers, where all the mathematical constants fit perfectly.

Maybe?

*edited for unclear language

Edited by Rabaelthazar
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when a korean is born he/she is 1!!???!! w?? by the way i dont give a fuk if there is a numbar between 0 and 1 <> Edited by bullit

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This might keep you busy trying to work it out: http://en.wikipedia.org/wiki/Principia_Mathematica

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So Peano naturally assumes that the sequence of integers goes something like

dude

dude plus one

etc

but can you actually PROVE that there is no integer between 0 and 1?

>>> I Don't think so!

Actually in this instance, there's 2 spaces and the letters 'and' between 0 and 1

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when a korean is born he/she is 1!!???!! w?? by the way i dont give a fuk if there is a numbar between 0 and 1 <> When a child is born it is already 9 months old. Is 9 months then a true "year" to the human biology, rather than a mere circling of the sun?

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^ I hope not! that means I'm over 50!!! ##### Share on other sites

The question itself is incompletely-defined. Excluded in the original post, are further rules to define the mathematics at hand (there are probably infinite mathematic rulesets). Is the set of numbers being constructed cyclic, or unbounded? If they're cyclic, then it's a simple matter to show that the cyclic group of numbers with modulus 2.5, and the construction N(n+1) = N(n) + 1 leads to a number between 0 and 1.
1, 2, 0.5 (=3 - 2.5)

This is not a contrived example, as much as it may seem. We use this type of modular arithmetic to tell the time. Once a time reaches 11:59, it wraps around to 0 again (0 being denoted by the number 12).

Now let's assume that we're not using a cyclic group - the group is unbounded. Then in this case, perhaps with a few additional rules (such as to define the notion of 'betweenness'), it is possible to prove there is no natural number between 0 and 1. Making a slapdash definition of betweeness as "A number X is between two other numbers Y and Z, if it greater than only one of Y and Z, and smaller than only the other". X is smaller than Y if X-Y is negative, and X is greater than Y if X-Y is positive.

Going by a mathematical induction route:
n = 2

2 - 1 = 1 => 2 is greater than one, 2 is not less than one
2 - 0 = 2 => 2 is greater than zero, 2 is not less than zero
Thus 2 is not between 0 and 1.

Assume that this is true for all positive n. Let's try and prove it for n + 1

n + 1 - 1 = n => n + 1 is greater than one, n + 1 is not less than one (as the remainder is n, and n is positive by definition)

n + 1 - 0 = n + 1 => n + 1 is greater than zero, n + 1 is not less than zero (as the remainder is n + 1, and it can be shown that n + 1 is positive too, due to the set being unbounded)

Thus for all positive integers >= 2, none are between 0 and 1.
It can be shown in a similar manner that going to negative numbers will not provide a solution either. So it we can then prove that the set of integers is all the numbers there are in such a system, then we have proved that there is no number between 0 and 1, in this particular system. I think it can be proven that the integers are all the numbers there are with a few more axioms that are usually implicit anyway.

So in short, it is the rules of the particular mathematic that determine whether such a question is true or not. There is no one mathematics, but a possibly infinite set of mathematical rulesets, some of which will contain a number between 0 and 1, and others which will not. It is similar to asking "does there exist a penis between the legs of humans" - for some humans it will be true, and for others it will not - we can't provide a binary answer to the question without further specifications. Edited by CβL
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