High quality Teaching such as presented by Professor Tao, makes the working of processes available to see for Students. From which the philosophical "Essence", the essential (amplitude and frequencies modulation) communication skills are interpreted and perceived. (Commentary) As yet another amateur (bridging) observation.., the frequent objective of math-phys-chem by philosophy is to unify Science in one word-concept, like "Universe", = one-turning, or Superspin connection etc, by following the vernacular of different fields and establishing from those path-aspects, a Correspondence in Principle. The reverse is true, (in principle of course), WYSIWYG in QM-TIMING substantiation, so the methodology of "proofing" in Mathematics that is all about improving techniques derived from empirical perceptions adapted to physical technology, eg after the Pure Mathematics, for Physics and Engineering, .. and can be observed to be self-defining, and so realised from the natural experiment of evolution, ..and then discovered by research. Therefore the common factor in Science is the "=" formulation, the "line of approach", observable identity, such as the axial-tangential visualisation of Polar-Cartesian Coordination in Perspective, ..of the coordinate zones "outside-inside" macro-micro formulae balanced at Zero. In which "outside" is the empty spacing of positioning excluded from rational/unitary modulated connection of e-Pi-i flat-space resonance, and "inside" is the Numerical superposition singularity of relative micro states/fractal of unity, but everything = one line of identification graphically/symbolically in the natural language of math-philosophy, ..quantized qualities of QM-Time. (= Almost incomprehensible simplicity of "All things Connected", taken for granted in developmental experience) Science and perceptions, Natural Philosophy, is Evolution and Life, (reiteration of identification imagery). "Discrete" is self-referencing connection, or reflection-imaging, single sided superposition, similar in concept to the characterization of "Majorana" a particular coordinate compound modular identity, in Continuity. Eternal continuous connection is the absolute Limit, or Potential Probability = 1, recognised in image-symbol components of reflection-resonance divisions/multiples, and balanced formulae.., = at Zero difference. The implication is that "zero or nothing" is the Real "inside-finite" limit-boundary of the unity-connection substantiation of existence in QM-TIMESPACE, in self-defining terms. Ie a real number exists in temporal quantum information terms as a fractional-fractal, dualistic algerbraic identity of unity and "image-inary" discrete dimensional numerical multiples. "Trivial" doesn't necessarily mean comprehensible(?).
Has anyone any idea who answered the question about ultraproducts by saying that he explained something to Tarski using them? I did not get it all, and I could not recognize his voice.
I think the person who talked about using ultraproducts to prove the compactness theorem in 1958 was likely Dana Scott, who was a student of Tarski's for a while, then switched to Alonzo Church. en.wikipedia.org/wiki/Dana_Scott
So prof Tao. As part of an on going discussion I just wanted to clarify that with the non standard creation of reals we can say as you have noted that the limit of 1/n as n goes infinity does not equal 1 (unlike the standard construction of reals where it is zero and you note no one really uses the standard construction of reals anymore). This leads me to note that with this inifintesmal between limit before zero we can also then say that 0.999... does not equal one in the non standard construction of reals (which include hyperreals ie infinitesmals).
That is , the limit of 1/n as n goes to infinity does not equal ZERO (typo previously) in the non standard construction of reals. Also 0.999... does not equal 1 in the non standard construction of reals. Unlike the standrad construction of reals.
***** He says it is important that the reals exists and not that you use that method of dealing with them as it is cumbersome. And he says that non-standard is an expansion of the reals meaning all truths in standard reals are equally true and valid in non-standard aka hyperreals. so 0,(9)=1 still,
I think, that prof Tao suppose that 0 is not natural number, 1/n where n is natural number has the limit, this limit is not natural number. For same people is not 0 natural number. Here in Czech Republic is 0 in theory sets, in mathematic analysis, but not natural number only for students younger 18 years (before then they go to University). We have very much differences, in CZ exists derivative when is limit infinity, in USA doesn't ....
i think math is more concerned with using discrete/linear structures to analyze continuous structures. why does he give no opposite direction examples?
this entire talk is about how to do that. the 'continuous counterexample' around the 55m-1h mark is an example of that technique. graph theory is a field hugely influenced by manifold theory, because manifolds are continuous graphs, made precise by the nonstd POV
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I have no idea what Terry is talking about, I just enjoy watching the supremely smart and humble do their work.
That's a good and coincise talk, really good lecture. I really like how he provides good enthusiasm of the topic beforehand.
Terence tao legend!!
Really wonderful!
I love the reference to approximate groups!!!
Outstanding!
Woah! I see shades of Aumann's agreement theorem in the x_n /elem X_n formal definition!
High quality Teaching such as presented by Professor Tao, makes the working of processes available to see for Students.
From which the philosophical "Essence", the essential (amplitude and frequencies modulation) communication skills are interpreted and perceived. (Commentary)
As yet another amateur (bridging) observation.., the frequent objective of math-phys-chem by philosophy is to unify Science in one word-concept, like "Universe", = one-turning, or Superspin connection etc, by following the vernacular of different fields and establishing from those path-aspects, a Correspondence in Principle.
The reverse is true, (in principle of course), WYSIWYG in QM-TIMING substantiation, so the methodology of "proofing" in Mathematics that is all about improving techniques derived from empirical perceptions adapted to physical technology, eg after the Pure Mathematics, for Physics and Engineering, .. and can be observed to be self-defining, and so realised from the natural experiment of evolution, ..and then discovered by research.
Therefore the common factor in Science is the "=" formulation, the "line of approach", observable identity, such as the axial-tangential visualisation of Polar-Cartesian Coordination in Perspective, ..of the coordinate zones "outside-inside" macro-micro formulae balanced at Zero.
In which "outside" is the empty spacing of positioning excluded from rational/unitary modulated connection of e-Pi-i flat-space resonance, and "inside" is the Numerical superposition singularity of relative micro states/fractal of unity, but everything = one line of identification graphically/symbolically in the natural language of math-philosophy, ..quantized qualities of QM-Time. (= Almost incomprehensible simplicity of "All things Connected", taken for granted in developmental experience)
Science and perceptions, Natural Philosophy, is Evolution and Life, (reiteration of identification imagery).
"Discrete" is self-referencing connection, or reflection-imaging, single sided superposition, similar in concept to the characterization of "Majorana" a particular coordinate compound modular identity, in Continuity.
Eternal continuous connection is the absolute Limit, or Potential Probability = 1, recognised in image-symbol components of reflection-resonance divisions/multiples, and balanced formulae.., = at Zero difference.
The implication is that "zero or nothing" is the Real "inside-finite" limit-boundary of the unity-connection substantiation of existence in QM-TIMESPACE, in self-defining terms. Ie a real number exists in temporal quantum information terms as a fractional-fractal, dualistic algerbraic identity of unity and "image-inary" discrete dimensional numerical multiples.
"Trivial" doesn't necessarily mean comprehensible(?).
I am reminded of the Banach-Tarski paradox...
My idol 😊
AHmazing ❤
awesome
Mr T of Math.
Why I think he was a student at the beginning?
Are we limited to monotonic sequences?
The analogue of Pontryagin duality for representations?
Has anyone any idea who answered the question about ultraproducts by saying that he explained something to Tarski using them? I did not get it all, and I could not recognize his voice.
I think the person who talked about using ultraproducts to prove the compactness theorem in 1958 was likely Dana Scott, who was a student of Tarski's for a while, then switched to Alonzo Church.
en.wikipedia.org/wiki/Dana_Scott
Did anybody get what he explained who Tarski?
So prof Tao. As part of an on going discussion I just wanted to clarify that with the non standard creation of reals we can say as you have noted that the limit of 1/n as n goes infinity does not equal 1 (unlike the standard construction of reals where it is zero and you note no one really uses the standard construction of reals anymore). This leads me to note that with this inifintesmal between limit before zero we can also then say that 0.999... does not equal one in the non standard construction of reals (which include hyperreals ie infinitesmals).
That is , the limit of 1/n as n goes to infinity does not equal ZERO (typo previously) in the non standard construction of reals. Also 0.999... does not equal 1 in the non standard construction of reals. Unlike the standrad construction of reals.
***** Where here?
***** He says it is important that the reals exists and not that you use that method of dealing with them as it is cumbersome.
And he says that non-standard is an expansion of the reals meaning all truths in standard reals are equally true and valid in non-standard aka hyperreals. so 0,(9)=1 still,
W
I think, that prof Tao suppose that 0 is not natural number, 1/n where n is natural number has the limit, this limit is not natural number. For same people is not 0 natural number. Here in Czech Republic is 0 in theory sets, in mathematic analysis, but not natural number only for students younger 18 years (before then they go to University). We have very much differences, in CZ exists derivative when is limit infinity, in USA doesn't ....
Aren't the hypereals the same as Conway's surreal numbers?
no their not. ur an idiot. hyper reals here mean non standards reals. surreals are a proper class, far bigger than hyper reals. u r a moron.
@@estring123 No need to call him a moron, moron.
No
yes, under NBG. see philip erlich's work on 'absolute arithmetic continuum'
@@soyoltoi
34:21
i think math is more concerned with using discrete/linear structures to analyze continuous structures. why does he give no opposite direction examples?
this entire talk is about how to do that. the 'continuous counterexample' around the 55m-1h mark is an example of that technique. graph theory is a field hugely influenced by manifold theory, because manifolds are continuous graphs, made precise by the nonstd POV
VERY DISTRACTING his speech pattern AND his facing the blackboard instead of audience.