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>define a set [math]^*\mathbb{R} = \mathbb{R} \cup \{\epsilon, -\epsilon\}[/math] that extends all the first-order operators and axioms (i.e. all axioms except the completeness axiom) as if [math]\epsilon, -\epsilon[/math] were real variables
>define an unsurprising axiom [math]-(\epsilon) = -\epsilon[/math]
>define a function [math]\mathbb{st}(^*x) = \lim_{\epsilon \to 0^+} \lim_{-\epsilon \to 0^-} {}^*x \in \mathbb{R}[/math]
Is this not an easy construction of the theory of the hyperreals? I'm thinking I must be missing something, since it wasn't particularly hard to come up with. This is essentially a formal summary of my rudimentary understanding of hyperreal numbers so bear with me if I've overlooked things. I can perform differentiation and take limits in general successfully and I find it much more intuitive with a better notation.
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>>8934272
Hint: 1+e is not in *R
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>>8934274
Read the first meme arrow more carefully.
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Why don't you even just read the Wikipedia page you mongoloid? No you are fucking wrong because the whole point of the exercise was to avoid using limits.
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>>8934280
This has nothing to do with [math](\epsilon, \delta)[/math] limits.
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>>8934282
Then what are those limits in your definition of the standard part?
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>>8934287
They're part of the definition of the standard part function, which is part of the theory of the hyperreal numbers. This is a construction of the hyperreal numbers that is a simple extension to the real numbers and is similar to the construction of the complex numbers in wide use today.
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>>8934294
Your official certificate of mental retardation should arrive in the mail in 4-6 weeks. Thanks for playing.
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>>8934297
Why am I retarded, then?
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NOTE: I normally think in terms of type theory. I am not nearly as familiar with set theory and may say strange things. I would have a much easier time writing this in pseudo-Agda.
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