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Topic: the irrelevant limit concept in Calculus, and why burden students
with nonsense? #22 Uni-text 8th ed.: TRUE CALCULUS; without the phony limit concept

Replies: 2   Last Post: Oct 22, 2013 9:34 PM

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plutonium.archimedes@gmail.com

Posts: 9,436
Registered: 3/31/08
people with fryed brains
Posted: Oct 22, 2013 9:34 PM
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On Tuesday, October 22, 2013 7:43:14 PM UTC-5, federat...@netzero.com wrote:
> On Tuesday, October 22, 2013 1:55:33 AM UTC-5, Archimedes Plutonium wrote:
>

> >Change in mathematics comes at a snail's pace, even when the establishment
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> >is grotesquely wrong, so I write this more for the present day teachers of
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> >Calculus, putting them to shame for the pain they inflict on young minds
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> >having to learn a pollution and not true calculus.
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> Okay. Fair enough. So, here's a challenge. Write down the axioms for Calculus. It should be powerful


Listen up bonehead, for somewhere in your education, your brains got fryed and that you should be in the insurance business or service industry and not mathematics or science. When your brains are fryed and cannot ever think clearly enough anymore, is the time to find work elsewhere.



enough to capture all the main results (e.g. Taylor's Theorem, the Mean Value Theorem) commonly seen in calculus treatments, as well as providing the basis for defining commonly-used functions such as the sine, logarithm, etc.
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> This task is not as easy as it sounds. All complete and consistent axiomatizations reside SOLELY in second order logic and must involve at least one second order statement (e.g. the Axiom of Completeness -- out of which the limit concept originates). A solely first-order axiomatization for Calculus must be either incomplete or inconsistent.
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> Incompleteness may not necessarily be a bad thing, if one admits models for the underlying theory other than the real number system. An example of this is the quasi-standard formulation of real numbers in a first order language, where "completeness" is replaced by a schema of first-order axioms.
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> Since such a system necessarily captures the second order theory of real numbers incompletely, then it must admit "non-standard" models -- out of which ultimately arise the theory of infinitesimals.
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> But all this is an aside: the task on hand is this -- write down a system of axioms, be it in first order logic or second order logic.
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> For reference: the "standard" formulation (which the term "non-standard" in "non-standard analysis" refers to) is the one contained in second order logic, which is:
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> (A) The real numbers comprise a field, with the operations a,b |-> a+b, a,b |-> ab respectively for addition and multiplication; the respective inverses a |-> -a and a != 0 |-> a^{-1}; and respective identities 0 and 1.
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> (B) The field is totally ordered by the relation a <= b (i.e. less than or equal). Ordering is such that the subset { x: x >= 0 } of non-negative numbers is closed under addition and multiplication.
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> (C) The ordering is complete.
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> Axiom (C) is the second-order statement, since it is actually a statement about statements (or equivalently: a statement about subsets).






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