Hi Billy T. ... Anyhow, since the discussion IS the question of the CURRENT AXIOMATIC SUFFICIENCY/DEFICIENCY when it comes TO trying to apply maths in reality contexts (such as the examples in post #146 and #152), then it is ENTIRELY PROPER that we examine those current axioms THEMSELVES, and NOT just keep repeating (like you are doing again) exercises and 'proofs' which FOLLOW those current axioms. ...

The ONLY improvements possible to existing axioms would be to remove any ambiguities, of which AFAIK there are none. You could remove an axiom and have different consequences follow, or add new ones. There is a whole lot of literature on the geometry that result when the axiom that parallel line do not intersect is removed to allow that they can.

With in the very useful normal set of axioms new concepts, like imaginary numbers, have been well defined and are quite useful in the real world, especially when describing the behavior of voltage and current relationship in circuits with reactance; however there is no claim within the mathematics that imaginary numbers will have any real world application. Some electrical engineers have discovered that they do (and that the response of linear circuits to complex wave forms is most easily achieved by expressing those forms in terms of their Fourier* components). Likewise quantum physics makes great use of mathematics, but mathematics it self is a CLOSED TAUTOLOGY and like all such

**closed** tautologies can make NO CLAIMS of application outside its closed domain. There are whole fields of mathematics that AFAIK are of interest only to mathematicians. (Perhaps parts of number theory are totally without any real world application, but large prime numbers are very valuable for making nearly unbreakable codes.)

If you apply some math to a real world problem and it is helpful, fine, but if in some situations doing that leads to a paradox or confusion,

**the fault is with you, not mathematics. ** Mathematicians warned you up front that it was a CLOSED TAUTOLOGY, based on an arbitrary set of postulated axioms. For example 41/333 is completely well defined both in the real world and in mathematics when also expressed as a decimal too. I. e. as the infinitely long sequence 0.123123123123... but the real world has problem with 41/333 expressed that way as all calculations are finite.

SUMMARY:

It is non-sense to speak of "CURRENT AXIOMATIC SUFFICIENCY/DEFICIENCY." (Sufficiency or deficiency for what?) Yes, the logical deductions from the current, most commonly postulated axioms can be misapplied.

All I have seen posted here faulting mathematics is ignorance and wasted time. If they don't like the results the current most common set of axiom produces, then suggest changes in the current set, either by adding new axioms or deleting one or more.

* Fourier died in 1830. AFAIK, his transforms had no real world application for ~100 years. Mathematician tend to be "Pencil & Paper" explorers. - That reminds me of old joke: University Administrator called head of Physics Department in complaining about their large budget request asking: "Why can't you be more like the Math department?" Physics head said: "We need lots of experimental equipment. All they need is paper and pencils and their greatest expense is for waste baskets."