i talk about epsilon delta arguments and theory of real analysis .It apparently seems that one does not use the fact that given 2 real numbers a and b either one is less than other or greater than other or they are equal is actually used in proof in real analysis and whenever it is used the use can be avoided or justified with the known real numbers or rational numbers.

The last upper bound axiom is not very intuitive., but cantor's nested interval property is more intuitive, this along with Archimedian property implies the least upper bound axiom. the acclimatization of real numbers in Diffidence's

treatise is does not use least upper bound axiom.

Dedkind cuts directive derive least upper bound. cauchy sequence construction is aesthetically less satisfying when compared to Dedkind.very recently there is another construction of R from the set N based on slopes.

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Brouwerian., Bishop constructive, and also recursive analysis have been developed over the past 100 years without the trichotomy law for the less than relation. Of course you can expect to lose as well as gain, what is lost is sharp theorems like the Brouer fixed point theorem or the Hahn Banach theorem, which hold in these contexts only with epsilon error. You can easily find the literature using google.

Thank u very much anil .nerode sir u are taliking of constructive real analysis. but what Hahn Banach guarantees is nonempty dual space and in any concrete example one can always establish It is only blanket theorems like Hahan banach theorem which pose a problem. same thing goes for existence of algebraic closure and existence of maximal ideals.

what I mean is only to avoid law of trichotomy but profs y contradiction will remain.

when i look at classical axiomitization of reals we specify a set of positive reals raher than specifying order.

if we look at real numbers a s binary strings it is very easy to distinguish a strictly positive number it is one i which one nonzero bit occurs . that may occur after n bits are read and n may vary. so negative nubers do not poseany problem testing wheher a real number is zero is not an innocent task

herein comes the relevance of Godel or halting problems because a computer program to test for a non zero bit may ave to run infinitely many steps in theory.

this philosophical point pinpoints the rather dubious existence of the set of Real numbers.

You have clearly not examined the literature I referred you to. The recursive analysis definition of a description of a real number is a pair consisting of a Cauchy sequence of rational numbers together with a rate of convergence function f(n). A rate of convergence function f is one such that to guarentee that terms of the rational sequence are within the reciprocal of n from each other, just go to terms beyond the f(n)'th6

A function G on the reals to the reals is called recursive if there is a Turing machine which, fed as input a description of real x, yields as output a description of G(x).

There is a Turing machine operating on descriptionsof reals r, s, which stops if and only if r is less than s.

There is noTuring machine which stops iff r=s. This corresponds to your intuition.

Concerning the Hahn-Banach theorem, long ago, in the 100th anniversary dedicatory volume to Brouwe6 about 1980, I and George Metakides published a paper giving an EXPLICIT counterexample to the Hahn-Banach theorem in recursive functional analysis. The related subject called reverse mathematics, pursued by Harvey Friedman and Steve Simpson and their school, determines the "level of non-recursiveness" of many, many mathematical propositions, and does provide EXPLICIT counterexamples if you understand the subject.

Brouwerian intuitionstic mathematics is analogous, but has no place for explicit counterexamples, It just shows that if there were constructions for such things, they would lead to contradictions, which is probably what you were referring to.

What i meant about Hahn Banach theorem is little different in context. given any concrete Banach space like euclidean plane or space of continuous functions it is easy to find nontrivial continuous linear functional s but to prove a blanket theorem like dual space is nonempty ( essentially Hanan Bamch theorem) requires axiom of choice. same thing given any field and a polynomial it is possible to have splitting field . but a blanket theorem like algebraic closure exists requires use of axiom of choice,

personally i find the axiom intuitive as Cartesian product of two sets is nonempty , three sets is bigger than two sets and so on..... the real point is in the definition of a family of sets which amounts to the abstract concept of a mapping

as a mathematician I find Turing machine unsatisfactory and too inelegant. the unlimited register machine is somewhat better idea.

but i do not understand why we neeed abstract machine. We can and should define the concepts of recursive function theory abstractly in terms of Zermelo frankel theory or intuitive set theory axiomatically as all other mathematical theories ..

unforunqtely if u see i put he idea in very simple erms using its 1 nd 0 and to determine ositive means here should be atleast one bit t turned on but we may have to wait potentially infinite readings to see that it is turned on .this is the cruxand everything else is verbiage .this is the simplest demonstration of the nebulousness of rreal numbers.

i do not mean big theories but simple ideas . all great innovations carry a simplicity with them

The Banach space in the paper referred to is a perfectly classical perfectly concrete recursively presented space. . There is no difference in results whatever formulation you use, Turing machines, or register machines or whatever. The concept of relative recursiveness, which is what is involved, is quite independent of which formulation you use.

Zermelo Frankel set theory is a superstructure which supports all the equivalent definitions of recuriveness. You should learn a little more about these subjects before making judgements about them. This concludes my remarks.

I beleive these notions are important only when they shed light on real numbers . intuiitve understanding which i remarked sheds enough light constructive real analysis a rg8 subject . y remarks about aha Banach theorem are very perfect reflected in many books . I request respected Prof Nerode to glance hroughprefaeof the classic foundations of modern analysis by J.Dieupdene. i beleive i alied computer science , compiler construction, or logic progrraming,, Analysis , but personally my tests do not incline to model theory or recursie functio theory as i feel their purpose is wel served intuitively . neither these theories are adding to applicatons of computer science (real world) or enrichment of Mathematics.

I Know turing machine , post machine URM are all equivalent but i feel one should list axioms or axiom schema for modelling computation rather than machines as hey fit more nicely into mathematical framework. this remark can be more appreciated by a mathematician than a computer scientist. my professor of computer science dr. S. K. Mehata when listening to my views remarked that ow can u model computation without a machine . Now that is a typical engineer or computer scientist mental framework. a mathematician would love it modeled axiomatically!.