Re: JSH: Logic and paradox

Jesse F. Hughes wrote:

Vladimir Dergachev <volodya@xxxxxxxxxxxxxx> writes:


Which shows once more that "=" does not denote an equation:

0 = O(X^n)
0 = O((x-1)^m)
=>? 0+0 = O(x^n)+O((x-1)^m) = nonsense
=>? 0+0 = nonsense


I disagree. The only property required of "=" is that if a=b and b=c then
a=c.

Wow. This is just wrong. Equality is an equivalence relation. Thus,
aside from the property you mention, we need symmetry (a = b implies
b = a) and reflexivity (a = a).

I readily concede this point.


But equality requires more than this: Two things are identical iff
they satisfy all the same formulas. In other words, a = b iff for all
formulas P, we have

P <-> P'

where P' is the result of replacing some free occurrences of a with b.

Yes, but what class of formulas are you talking about ?

No offence, but even in logic you need context, you can't just take notation
and substitute away. It might be very little context like "integer
arithmetic theory", but still context.

I usually read a=b as a is equivalent to b using the equivalence relation
natural to objects a and b.

So 10=3 if 10 and 3 are considered part of Z_7 and I do not think it is
wrong that 10!<> 3! (even in Z_7).

The big-Oh notation is a horrible abuse of notation, since it misuses
the equality symbol. I wonder who first used it? It's such a strange
use that it reminds me of the terrible slang these days, when people
say "This equals that" but mean "This produces that."

I prefer => symbol myself (lets me know when I am dealing with logic
statements and when with mathematical objects), but I do think O(x)
notation to be very useful, convenient - and, if you follow the pencil
algebra definition, perfectly rigorous and clear.

best

Vladimir Dergachev


.

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