Time for another round of examining the interesting beliefs of SmartmanApps and #debunking the #disinformation this #MathsMonday.
We saw last time that his view of mathematics is at odds with that of the mainstream: I enumerated the standard axioms of the real numbers and proved that there can be no number 0.999β¦ that is simultaneously less than 1 and greater than 0.9, 0.99, and any such finite decimal truncation of 0.999β¦.
His idea is that 1 is βthe limit ofβ 0.999β¦, but not the exact value of it. But what exactly is a limit? Letβs have a look at what our Smart Friend calls a limit:
> The limit is the number which [the sequence] never reaches
(see https://dotnet.social/@SmartmanApps/116303201093245275 I am not quite certain he intends this language to apply to all sequences, but I have not seen other descriptions from him)
This is simply inadequate, and itβs worth seeing why such poor explanations are inadequate with some examples.
* The sequence (1, 1, 1, β¦) *never reaches* the number 2, so is 2 the limit? The same applies for every number greater than 1, so are all of them the limit? The wording βthe numberβ implies that the limit should be unique.
* On the other hand, it *does* reach 1, which intuition says ought to be the limit of this sequence.
* The sequence (1, 2, 3, 4, β¦) will exceed every number eventually and so, I guess, βreachesβ every number. Again the wording βthe numberβ suggests that such a number always exists, but apparently does not.
It may be that the Genius has a more precise idea of limit lurking in his mind, but to tease it out weβd have to interrogate him about these (and probably other) examples, and most likely anyone who tried would get blocked before they could complete their investigation.
The usual definition can be seen clearly from the early 19th century, due to Bolzano, though its roots go back further. That definition is:
> For a sequence (a_0, a_1, a_2, β¦), and a real number A, if for every real number Ξ΅ > 0, there is some natural number N such that for every n > N, |a_n - A| < Ξ΅, then we say that A is the limit of the sequence (a_0, a_1, a_2, β¦).
This is quite a mouthful, and first year mathematics undergraduates spend quite some time getting the hang of it. A characteristic of the definition is the alternating *quantifiers*, which are written out βfor everyβ and βthere isβ here (but would normally be written with symbols). It took mathematicians some time to come up with this modern version of quantified mathematical language.
Nevertheless we can put it into simpler language, at the cost of a little precision: **the limit of a sequence is A if the sequence gets as close to A as we like and remains that close forever**. Itβs important that we keep that βremains that closeβ in. Itβs important that neither of these ways of describing the concept assume a limit exists, because not all sequences have a limit. It is quite easy to prove directly from the definition and the properties of the real numbers that:
* A constant sequence (a, a, a, β¦) has a as its limit
* If a sequence has a limit, the limit is unique
* The sequence (1, 2, 3, β¦) does not have a limit
The ordinary way of proving such basic facts is via our friends Completeness and the Archimedean property. Our pal has explicitly rejected these (by affirming the existence of infinitesimals) and so does not have them available for this purpose.
To see this practically, how should we prove that the limit of the sequence (0.9, 0.99, 0.999, β¦) is 1? The ordinary way would be to appeal to the definition:
1. Pick any positive distance Ξ΅. By the Archimedean property, Ξ΅ > 1/N > 1/10^N for some N
2. If n > N then 1-0.99β¦99 (with n nines) is less than 1/10^N < Ξ΅, so 1 is the limit.
The astute reader will notice this argument is very similar to the one from last week. But if infinitesimals exist, we *cannot do this*: the first step is, in fact, false. If Ξ΅ is infinitesimal, then there will not be any N such that Ξ΅ > 1/N! That is in fact what it means to be infinitesimal!
Specifically, if Ξ΅ = 1 - 0.999β¦, which the Smart Man says is greater than zero, this argument falls down; we cannot get the sequence (0.9, 0.99, 0.999, β¦) to be Ξ΅-close to 1 if Ξ΅ is infinitesimal by looking to some point far enough into the sequence: for any n, the nth term 1/10^n away from 1, and 1/10^n is larger than Ξ΅.
From further reading of SmartmanAppsβ posts, I suspect he might want to object that if we continue the sequence βto infinityβ the difference becomes infinitesimal. I should be very clear here: sequences as here defined and as used by him cannot be continued βto infinityβ. The defining rule for this sequence is that the nth term is 1 - 1/10^n, something which makes sense and is defined for *natural numbers* n, and because infinity is not a natural number and 10^β is not defined, we canβt just continue like that. The only way would be to make a *definition* of what 1 - 1/10^β means, i.e. to *choose* what happens at this continuation; there are no axioms governing rational numbers that force us to give a certain value to this expression.
Another potential objection is that I have used the βwrongβ definition of a limit, but:
1. You can find this definition in every single textbook and set of lecture notes on real analysis
2. You can find this definition (written in an old fashioned way of "variables that take on successive values" rather than sequences) in the 120-year old algebra textbooks he loves to cite
3. We saw multiple problems with his broken pseudo-definition that make it useless
so itβs up to him to provide a correct one. One could try as a first attempt to replace βfor every real number greater than zeroβ with βfor every non-infinitesimal real number greater than zeroβ. But without Completeness, basic facts like the uniqueness of limits, on which many more important theorems rest, would still be false.
Itβs fun to explore what happens to mathematics if throw out some of its founding principles, though it does make doing anything useful with it hard. Forget working out anything truly useful like calculus without Completeness, or something precise to replace it!
Next time I plan to look a bit more at infinitesimals and how you can treat them rigorously.
#maths #math #mathematics
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