Re: Cantorian pseudomathematics

David R Tribble wrote:
>> Infinitesimals don't sum up to any finite quantity. If they did, they
>> would not be infinitesimal.
>

Ross A. Finlayson wrote:
>> No, they do and they are.
>> Consider epsilon-delta, limit, and the integration of constant 1 dx
>> over zero to one. Is that not the summation of infinitely many
>> infinitely small areas to get a very known quantity?
>

David R Tribble wrote:
>> Uncountably infinitely many.
>> But a countably infinite sum of infinitesimals sums to zero.
>>
>> That's why you use
>> integral{0 to oo} f(x) dx
>> instead of
>> sum{0 to oo} f(x_i)
>>
>> The sum is countable, the integral is not.
>

Ross A. Finlayson wrote:
> Ever notice how the integral bar looks like an S? It's S for
> Summation, in the, as we were discussing, infinitesimal analysis,
> useful and used for hundreds and hundreds of years.

Ever notice that there are two kinds of sums in calculus? There's the
sigma notation for countable (discrete) sums, and then there's the
curly-S notation for uncountable (continuous) sums, a.k.a. integrals
or antiderivatives.


> Then there was the
> period about the rigorousness (rigorosity) of infinitesimals, and the
> limit was found to be a useful tool in establishing an inductive
> guarantee of these types of notions as Sum n=1^oo 1/n = 1.
>
> What would you say is dx?

An infinitesimal. And any integral expression involving dx uses an
uncountable infinity of them, just like I said before.

But a countable sum of infinitesimals is always zero, just like I
said before.


[Obligatory "there is no universe in ZF" text snipped.]

.

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