# Is there a paradox of third-order arithmetic?

Calculus, sometimes analysis or second-order arithmetic, seems more intuitive when formulated in infinitesimal terms than in terms of real-valued limits. However, the meta-theory of analysis, i.e. its rigorization, seems more intuitively explainable by the implementation in limits than by the implementation of infinitesimals (here implementation concerns semiotics as logically intertwined with syntax and semantics). Robinson's deduction of infinitesimals requires the elaborate machinery of model theory to get off the ground, whereas the calculus of limits was developed without recourse to, because historically preceding, model theory. Definitions from smooth infinitesimal analysis are relatively counterintuitive, too, at least insofar as LEM-suspension seems counterintuitive.

Is there some reason why first-order infinitesimal analysis would be intuitive whereas second-order limit analysis would be intuitive, such that second-order infinitesimal analysis reciprocates the counterintuitive aspect of first-order limit analysis? I'm imagining a dialectical moment where the proponents of infinitesimal analysis are in a dead-lock with limit-theory proponents because neither has yet shown which framework is more intuitive in fourth-order arithmetic, or worse, fourth-order arithmetic is beyond intuition anyway and so our conflicting second- and third-order intuitions have no higher court to appeal to, here.

Alternate formulation: some meta-theories of infinitesimals are themselves more intuitive than others. For example, again, smooth infinitesimal analysis is LEM-divergent and so is less intuitive than the classical limit theory, but it is essentially more intuitive, because intuitionism-adjacent, than Robinson's model-theoretic justification (perhaps). Or there are perhaps incommensurable degrees of intuition respecting different meta-theories, e.g. if Conway infinitesimals (transfinite surreal ratios) are intuitive enough for how intuitive the birthday system is, and are neither externally more nor less intuitive than some other meta-theory's harvest.

One question we can quickly ask, then, is whether the internal antinomy of infinitesimal theory has a solution as an indication that there is something inherently incomplete about our range of theories about infinitesimals. That is, we would note that none of the known theories is absolutely intuitive (no matter what it calls itself) or at any rate dominantly intuitive compared to all others. So we would infer that there must be at least one further such theory that we haven't come up with, and perhaps we could infer that there are indefinitely many possible further theories, here.

If the paradox of intuitions about infinitesimal theories has an internal solution, does this solution justify our belief in the external appearance of a specific "paradox of third-order arithmetic," which is then an antinomy of the Continuum, too? In this case, we have the thesis as one introduction of infinitesimals and the antithesis as some other, by the most direct possible route:

1. The nilpotence definition: a2 = 0, for infinitesimals a.
2. The reciprocal definition: n/X is an infinitesimal for transfinite X.

The dialectic can be framed as between nilpotence and idempotenceYYY... on the Kant set (or a Kantian space), then, where the justification of the reciprocal definition comes out of a justification of the idempotence thesis modulo division properties shared with 0, where there are types of infinitesimals for every combination of shareable properties as such. So this would seem like an indication of the "yet-to-be-discovered/invented theories" problem in that infinitesimals seem to lack such descriptions in texts about infinitesimal meta-theory up to this time (at least, I'm not seeing the simplest example of this introduction method in any of the calculus/related texts I'm looking through, although one of the methods I am seeing might cover the example and I'm just not understanding the notation, there).

YYY...That is, (1/X)2 is either idempotent or surreal, but is never nilpotent. Ironically, the sense in which nilpotence and idempotence are reciprocal concepts lends itself to some justification for both nilpotent and idempotent infinitesimals, even though the latter tend more towards Platonic realism than the former (so the intuitionist would tend to favor nilpotent infinitesimals alone, then).

• "I don't know such stuff." :-) Jul 22, 2023 at 14:18
• “the meta-theory of analysis, i.e. its rigorization, seems more intuitively explainable by the implementation in limits than by the implementation of infinitesimals” – please explain what you mean by this. Feb 21 at 5:11
• There seems some paradox of third order arithmetic since there seems no extra conceivable application utility for any of these ultra categorical third order arithmetics compared to Simpson's second order reverse engineered arithmetics... Feb 21 at 6:49
• Curse you, Community Bot, for bumping this! @JustSomeOldMan what I meant was that it is easier to understand the theory of the theory of limits than to understand the model theory that initially grounded Robinson's infinitesimals. The meta-theory of Conway's infinitesimals is yet easier to understand, and there are other groundings now of Robinson's too, so as I have learned (a little bit) more about this topic, I should have to revise my confusion over an alleged "paradox" of intuitiveness, here. Feb 21 at 11:12
• @KristianBerry But isn’t intuitiveness to a large degree subjective? I know it is especially so with regard to nonstandard vs. standard arithmetic. For example, I have heard of much controversy of the intuitiveness of infinitesimals for any arithmetic or theory of any order because those mathematicians find the standard approach more intuitive across the board. Feb 21 at 16:37