A Conversation for Mathematics

Russell's Paradox (antinomy)

Post 1


You state in the article that to get round the problem of Russell's paradox mathematicians "declared such questions illegal" (or words to that effect). While this is true in some sense it can be a bit misleading. It's not that mathematicians couldn't cope with the 'paradox' and tried to sweep it under the carpet by banning any questions which caused problem; but rather they realised that the previous version of set theory being used (known as Naive Set Theory) wasn't good enough, and a far more rigorous axiomatic system, Axiomatic Set Theory, had to be developed in which such questions could not be posed (i.e. it didn't make logical sense to talk of a set containing an infinite number of sets).

This can get a bit deep and there's no need to go into it here, but I thought it worth clarifying that declaring such questions illegal was more a matter of mathematicians correctly their existing theory, rather than cheating.

Russell's Paradox (antinomy)

Post 2


Posting to move to Brunel.

Pliny ate my posting.

Perhaps next time I get a dog I will name the brute 'Pliny'.

Russell's Paradox (antinomy)

Post 3


...like dividing by zero being an illegal opereration.

If 0/0=0, this represents a violation of the commutativity law, "where if n=n, n/n=1", so the commutativity law becomes, "where 'n' is not equal to zero and n=n, n/n=1".

If 0=0'=zero and 0/0'=1, then 1x0=0 (true) and 1x0'=1 (contradiction).

0/1 is not an illegal operation, 0/1=0, 0x0=0 and 0x1=0, all statements true and correct without contradiction.

On the other hand, if 1/0=0, then 0x1=0 (true) and 0x0=1, (contradiction).

If 0x0=1 were true, one could easily prove any number to be equal to any other number.

Meanwhile, there's a semantics issue.

In mathematical jargon, the terms 'outlawed' and 'contradiction' mean substantially the same thing, just as they do on a basis of literal interpretation of their roots.

If a statement embodies a 'contradiction', is, in other words, false to fact, it is 'outlawed' on that account.

There's nothing arbitrary about it, the process follows automatically from the laws governing mathematics and conformity to fact.

Russell's Paradox (antinomy)

Post 4



There is a lesser sense of the term 'contradiction' which means, merely, 'error'.

The term 'outlawed' is usually reserved for cases of violation of axiom.

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