Reverse Mathematics coming to the masses?

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Mostowski Collapse

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Sep 13, 2021, 7:30:03 PMSep 13
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Reverse Mathematics: Proofs from the Inside Out
John Stillwell - 2018
https://www.amazon.com/dp/0691177171

Mostowski Collapse

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Sep 13, 2021, 8:30:36 PMSep 13
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A review intro:

"Reverse mathematics is a programme in mathematical logic,
initiated in the mid-1970s, which seeks to determine which
axioms are necessary to prove theorems in areas of ordinary
mathematics such as real analysis, countable abstract algebra,
countably infinite combinatorics, and the topology of complete
separable metric spaces. Reverse Mathematics: Proofs From
the Inside Out is the first popular book on the subject, aimed at
advanced undergraduates in mathematics, but also a good introduction
for philosophers of mathematics. The time is certainly ripe for such
a book, bringing this fascinating area of contemporary mathematical
logic to a broader audience."

Review of John Stillwell,
Reverse Mathematics: Proofs from the Inside Out
Benedict Eastaugh∗ bene...@eastaugh.net
22 November, 2019
https://extralogical.net/files/stillwell-review.pdf

Khong Dong

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Sep 13, 2021, 9:03:50 PMSep 13
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On Monday, 13 September 2021 at 18:30:36 UTC-6, Mostowski Collapse wrote:
> A review intro:
>
> "Reverse mathematics is a programme in mathematical logic,
> initiated in the mid-1970s, which seeks to determine which
> axioms are necessary to prove theorems in areas of ordinary
> mathematics such as real analysis, countable abstract algebra,
> countably infinite combinatorics, and the topology of complete
> separable metric spaces. Reverse Mathematics: Proofs From
> the Inside Out is the first popular book on the subject, aimed at
> advanced undergraduates in mathematics, but also a good introduction
> for philosophers of mathematics. The time is certainly ripe for such
> a book, bringing this fascinating area of contemporary mathematical
> logic to a broader audience."

This statement "which seeks to determine which axioms are necessary to prove theorems in areas of ordinary mathematics such as real analysis" is a tall order and dooms the programme in its inception.

The reason being is "ordinary mathematics" is fatally faulty with Plato/Tarski/Gödel reliance on unsound _semantical_ "truths" that actually don't correspond with syntactical _provability_ facts.

Khong Dong

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Sep 13, 2021, 9:10:37 PMSep 13
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On Monday, 13 September 2021 at 19:03:50 UTC-6, Khong Dong wrote:
> On Monday, 13 September 2021 at 18:30:36 UTC-6, Mostowski Collapse wrote:
> > A review intro:
> >
> > "Reverse mathematics is a programme in mathematical logic,
> > initiated in the mid-1970s, which seeks to determine which
> > axioms are necessary to prove theorems in areas of ordinary
> > mathematics such as real analysis, countable abstract algebra,
> > countably infinite combinatorics, and the topology of complete
> > separable metric spaces. Reverse Mathematics: Proofs From
> > the Inside Out is the first popular book on the subject, aimed at
> > advanced undergraduates in mathematics, but also a good introduction
> > for philosophers of mathematics. The time is certainly ripe for such
> > a book, bringing this fascinating area of contemporary mathematical
> > logic to a broader audience."
> This statement "which seeks to determine which axioms are necessary to prove theorems in areas of ordinary mathematics such as real analysis" is a tall order and dooms the programme in its inception.
>
> The reason being is "ordinary mathematics" is fatally faulty with Plato/Tarski/Gödel reliance on unsound _semantical_ "truths" that actually don't correspond with syntactical _provability_ facts.

So there can't be effective or even valid ways in general "to determine which axioms are necessary to prove theorems".

Ordinary mathematicians (most of them) are just keep dreaming. All they'd need to understand is just looking at the axiom which is the negation of Shoenfield's N8 axiom and see if they can make sense out of it -- or out of its subsequent theorems!

Fritz Feldhase

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Sep 13, 2021, 9:15:38 PMSep 13
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On Tuesday, September 14, 2021 at 3:03:50 AM UTC+2, khongdo...@gmail.com wrote:

> "ordinary mathematics" is fatally faulty with Plato/Tarski/Gödel reliance on [etc.]

Please don't forget about Cantor, i. e. set theory!

Khong Dong

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Sep 13, 2021, 9:45:08 PMSep 13
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Oh no, you're just mistaken. The essence of Cantor theorem is valid, it's just the way failed mathematicians have waxed semantical "cardinality" over the essence that is wrong.

The essence of Cantor theorem basically asserts that, to the extend semantical interpretation is desired, a function is homological with a relation but not necessarily the other way around!

Khong Dong

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Sep 13, 2021, 10:39:21 PMSep 13
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Even just on syntactical basis, the essence of Cantor theorem basically asserts that:

invalid((y = F(x1, ..., xn)) ↔ R(x1, ..., xn, y))

without "cardinality"-semantical-waxing being remotely necessary at all.

Khong Dong

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Sep 14, 2021, 12:40:10 AMSep 14
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where, mind you, 'F' and 'R' would count as one non-logical symbol.

Mostowski Collapse

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Sep 14, 2021, 2:36:50 AMSep 14
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You mean like the Ding Dong theorem prime numbers that
are not prime number. Thats indeed a fault, in your brain.
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