The first presentation of Gödel's incompleteness theorem that I read through was in the back of a logic puzzle book by Raymond Smullyan which I enjoyed: The Lady or the Tiger?: and Other Logic Puzzles
To my understanding incompleteness theorem applies generally to bottom up constructions of naturals by additive algorithm. On the other hand, how does Gödel's proof relate to top down construction of number theory by nesting algorithm, defining first fractions and then integers and naturals as mereological decompositions of fractions?
@@hungrymathprof By concatenating mediants first in primitive operator language: < > < > < < > etc. For tally operation, define that < and > are numerator elements with value 1/0 and is the denominator element with value 0/1. Thus the second row has numerical interpretation: 1/0 0/1 1/01 As we see, that is a two-sided Stern-Brocot type generator, and tally of the generated words so that < and > are not counted as numerator elements when they are reserved by the denominator element confirmes that we get totally ordered coprime fractions. For example on the next row we get mediant word
Interesting, well I guess we have to ask the question: In what system do you propose doing this construction? If you propose doing this construction in set theory, then Godel's theorem should apply. If you propose doing this construction in some other foundational system, then you must describe the foundational framework you want to use to ground your model. Unless you just want to describe a different way of representing fractions, in which case Godel's theorem is irrelevant.
@@hungrymathprof This is intended for a constructivist foundational language grounded in holistic mereology, which encompasses both the primary holistic perspective (top down nesting algorithms) as well as reductionistic perspective (bottom up additive algorithms). The operators < and > symbolize continuous directed movement as the ontological primitive, so they can be interpreted as arrows of time, object independent relational operators, general recursion etc., as well as the Dyck language pair at the top of the hyperoperation tower, giving concrete intuitive explanation e.g. for the field arithmetic order of operations: first parentheses, then hyperoperations in descending logarithmic order. Inverse Dyck pair > < is also included as generator, and has also very interesting properties, and we can further construct full combinatorics of chirally symmetric generator rows. Lot to study. :) I've been slow by slow developing an idea of starting math from a "clean slate" as my foundational hobby, but recently I found out that philosophically this is not a new idea, but basically the same old foundational view of Plato's Academy, as presented in Proclus' commentary to First book of Elementa. There has been also many major steps forwards since Euclid in the creative front of constructive mathematics, Stern-Brocot, origami method, the undecidability proofs by Gödel, Church, Turing and others, which show the limits of strictly bivalent logic as a proponent for a foundational theory. I go with Coherence theory of truth as the source of mathematical truth, and. To my understanding this means that for foundational coherence the operators e.g. with semantics < 'increasing' and > 'decreasing' need to be bounded by the Halting problem on the most general level. Philosophically coherent "incompleteness" was presented already by Brouwer's pre-linguistic ontology of mathematics. What happens if an analogy of Gödel's technical proof could be constructed in this language, that I can't say. The only meaningful philosophical difference between Brouwer's Intuitionism and Gödel's timeless Platonism is that Brouwer has temporal ontology and Gödel non-temporal. Two-directional quantum time as cosmic mathematical times is a nice synthesis that IMHO solves this difference of opinion.
@@hungrymathprof It is a rare pleasure to meet a fellow traveller with a foundational passion. I hope that I did not bore you to death with my (hopefolly not too incoherent) blather. I just happen to think that the way forward after Gödel, Church, Turing etc. is to make our math more complete by accepting incompleteness theorem etc. undecidability of the Halting problem as fundamental to our temporal views of mathematical ontology. I just listened a very remarkable and hope nurturing dialogue between Curt Jaimungal and Mathew Segall (with title "The Conscious Cosmos" if you wanna search it). In terms of recent history of mathematics, the key quote from that discussion was: After Gödel debunked Hilbert's program including Principia Mathematica as the frontrunner, "Russel was devastated, Whitehead felt revealed". Russel passed his torch to Wittgenstein, who has been philosophically the most important philosophical critic of the direction of linguistic turn of Cantorism (and by extension Hilbertism). Whitehead progressed into more creative approach of process ontology, which in my view is essentially return to original Planonism as mathematically developed in the grove of Akademeia, with master mathematician Eudoxus as the second scholarchos of the Academy. BTW the original Greek term in Elementa that has been translated into English as "proportion" is analogy. My own background is Greek philology, and Euclid's term family logos (ratio), analogos (proportion) and alogos (irrational) is very essentially an organic whole, and Heath's translations don't do a good job in translating the original organic meaning. I don't know how familiar Whitehead was with the original Greek of Elementa, but nevertheless post PM he more or less returned to the old Greek paradigm of the original Academy with some cumulative novel perspectives.
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The first presentation of Gödel's incompleteness theorem that I read through was in the back of a logic puzzle book by Raymond Smullyan which I enjoyed: The Lady or the Tiger?: and Other Logic Puzzles
Oh yeah, Smullyan is fantastic! I will have to check that one out.
To my understanding incompleteness theorem applies generally to bottom up constructions of naturals by additive algorithm. On the other hand, how does Gödel's proof relate to top down construction of number theory by nesting algorithm, defining first fractions and then integers and naturals as mereological decompositions of fractions?
How do you propose to define the fractions first?
@@hungrymathprof By concatenating mediants first in primitive operator language:
< >
< >
<
< >
etc.
For tally operation, define that < and > are numerator elements with value 1/0 and is the denominator element with value 0/1.
Thus the second row has numerical interpretation:
1/0 0/1 1/01
As we see, that is a two-sided Stern-Brocot type generator, and tally of the generated words so that < and > are not counted as numerator elements when they are reserved by the denominator element confirmes that we get totally ordered coprime fractions.
For example on the next row we get mediant word
Interesting, well I guess we have to ask the question: In what system do you propose doing this construction? If you propose doing this construction in set theory, then Godel's theorem should apply. If you propose doing this construction in some other foundational system, then you must describe the foundational framework you want to use to ground your model. Unless you just want to describe a different way of representing fractions, in which case Godel's theorem is irrelevant.
@@hungrymathprof This is intended for a constructivist foundational language grounded in holistic mereology, which encompasses both the primary holistic perspective (top down nesting algorithms) as well as reductionistic perspective (bottom up additive algorithms).
The operators < and > symbolize continuous directed movement as the ontological primitive, so they can be interpreted as arrows of time, object independent relational operators, general recursion etc., as well as the Dyck language pair at the top of the hyperoperation tower, giving concrete intuitive explanation e.g. for the field arithmetic order of operations: first parentheses, then hyperoperations in descending logarithmic order. Inverse Dyck pair > < is also included as generator, and has also very interesting properties, and we can further construct full combinatorics of chirally symmetric generator rows. Lot to study. :)
I've been slow by slow developing an idea of starting math from a "clean slate" as my foundational hobby, but recently I found out that philosophically this is not a new idea, but basically the same old foundational view of Plato's Academy, as presented in Proclus' commentary to First book of Elementa. There has been also many major steps forwards since Euclid in the creative front of constructive mathematics, Stern-Brocot, origami method, the undecidability proofs by Gödel, Church, Turing and others, which show the limits of strictly bivalent logic as a proponent for a foundational theory.
I go with Coherence theory of truth as the source of mathematical truth, and. To my understanding this means that for foundational coherence the operators e.g. with semantics < 'increasing' and > 'decreasing' need to be bounded by the Halting problem on the most general level.
Philosophically coherent "incompleteness" was presented already by Brouwer's pre-linguistic ontology of mathematics. What happens if an analogy of Gödel's technical proof could be constructed in this language, that I can't say.
The only meaningful philosophical difference between Brouwer's Intuitionism and Gödel's timeless Platonism is that Brouwer has temporal ontology and Gödel non-temporal. Two-directional quantum time as cosmic mathematical times is a nice synthesis that IMHO solves this difference of opinion.
@@hungrymathprof It is a rare pleasure to meet a fellow traveller with a foundational passion. I hope that I did not bore you to death with my (hopefolly not too incoherent) blather. I just happen to think that the way forward after Gödel, Church, Turing etc. is to make our math more complete by accepting incompleteness theorem etc. undecidability of the Halting problem as fundamental to our temporal views of mathematical ontology.
I just listened a very remarkable and hope nurturing dialogue between Curt Jaimungal and Mathew Segall (with title "The Conscious Cosmos" if you wanna search it). In terms of recent history of mathematics, the key quote from that discussion was:
After Gödel debunked Hilbert's program including Principia Mathematica as the frontrunner, "Russel was devastated, Whitehead felt revealed". Russel passed his torch to Wittgenstein, who has been philosophically the most important philosophical critic of the direction of linguistic turn of Cantorism (and by extension Hilbertism).
Whitehead progressed into more creative approach of process ontology, which in my view is essentially return to original Planonism as mathematically developed in the grove of Akademeia, with master mathematician Eudoxus as the second scholarchos of the Academy.
BTW the original Greek term in Elementa that has been translated into English as "proportion" is analogy.
My own background is Greek philology, and Euclid's term family logos (ratio), analogos (proportion) and alogos (irrational) is very essentially an organic whole, and Heath's translations don't do a good job in translating the original organic meaning.
I don't know how familiar Whitehead was with the original Greek of Elementa, but nevertheless post PM he more or less returned to the old Greek paradigm of the original Academy with some cumulative novel perspectives.