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Is a mathematical zero impossible?


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This suggests that the idea of the number zero or the empty set can be convenient, or useful, but not necessarily meaningful. An example of an empty product is given as the factorial, 0! = 1.

A good example of where the empty set comes up in the the context of the solution set of a system of linear equations. A system that is inconsistent has the empty set as its solution set.

 

I am not sure what you really mean by meaningful. The meaning depends on the context for sure.

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A good example of where the empty set comes up in the the context of the solution set of a system of linear equations. A system that is inconsistent has the empty set as its solution set.

 

I am not sure what you really mean by meaningful. The meaning depends on the context for sure.

 

Some examples in that Wikipedia article don't appear to be problematic - not, to me anyway. For example, I see no problem with x0 = 1. However, the "empty product" 0! = 1 caught my eye. (The article was about empty products)

 

n! is usually defined as n! = n x (n-2) x (n-3) x .... 2 x 1. By "meaningful" I was referring to the fact that it isn't obvious (not to me anyway) how 0! = 1 can be reconciled with that. The article did refer to equating such products to 1 as a convention.

Edited by JonG
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However, the "empty product" 0! = 1 caught my eye.

I think that is usually taken as a definition, largely because it seems to fit well with the combinatorial formula that involve the factorial. For example, the number of ways of choosing k things from a collection of n things is n!/k!(n-k)! Now what about the situation where n = k = 2?

 

You are forced to think about 1/0! = 1 as there is only one way you can chose 2 things from a collection of two things. And so 0! = 1, by definition.

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For example, the number of ways of choosing k things from a collection of n things is n!/k!(n-k)! Now what about the situation where n = k = 2?

 

Yes, that sort of argument seems reasonable. Another simple one is n!/n = (n - 1)! and then put n = 1.

 

However it still seems odd that a factorial, which is normally defined as a product of terms, can be identified with 0!. But I suppose that it isn't any more peculiar than x0 = 1.

 

Anyway, my point isn't relevant to the original post - apologies for the diversion, and thank you for your comments.

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It is curious that one way to construct a number system starts with nothing and generates a set with repeated copies as members to generate the natural numbers in the process.

 

Another interesting observation on the empty set is that 'nothing' distinguishes the two basic objects in geometrical topology (of 3D), the sphere and the torus.

 

The 'hole' is made of 'nothing', but unless it exists the object is a sphere, not a torus

 

So mathematics is founded at least in part, on nothing at all!

Edited by studiot
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