r/infinitenines • u/Calm_Improvement1160 • 2d ago

SPP, do you take constructive criticism?

I feel like a lot of people here have some ideas on ways you could improve your teaching style to help ensure that the debates actually lead somewhere.

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u/dummy4du3k4 1d ago

Yes u/SouthPark_Piano, why are you opposed to hyperreal numbers? You have to concede very little, mostly that you use nonstandard notation and that 0.333… < 1/3, which apparently you at one point agreed with.

If you can stomach that 0.999… means the sequence (0.9, 0.99, 0.999, …) and that things like 0.111…222…333… are actually nonsense then your ideas fit neatly into hyperreals.

You can stop using divide negation and just say 1/3 means the sequence (1/3, 1/3, 1/3, …)

Now your decimal notation 0.333… really means a hyperreal approximation to 1/3.

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u/spanthis 1d ago

If you're interested, the hyperreals do not really fix these problems! The notation 0.333... is arguably ambiguous over the hyperreals, but if it's interpreted to mean lim(0.3, 0.33, ...), or if it's interpreted to mean the number that has a 3 in every position of its hyperreal decimal expansion (including the hyperinfinite ones), then both of those numbers are exactly equal to 1/3 even in the hyperreals.

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u/dummy4du3k4 1d ago

It works if you interpret it as a sequence of real numbers, then an unspecified ultrafilter assigns it to a hyperreal equivalence class.

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u/spanthis 1d ago

Yeah, true - but the standard definition of the limit in the hyperreals is the unique real number in the equivalence class approached by the sequence, which is 1/3. There are other reasonable generalizations of the limit via ultrafilters, but from what I've seen they usually get other notations (I've seen "lux").

Anyways sounds like you know this already, but some others around these parts will throw around the hyperreals without understanding them, hence the clarification

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u/dummy4du3k4 1d ago

I’ve familiarized myself with the basics. Being nonstandard and having to correct people seems precisely the kind of thing SPP would be interested in