Cantor's diagonal argument says that any list of reals is incomplete. We can prove it directly by showing that any list of reals (not an assumed complete list, just any arbitary list) is necessarily missing the antidiagonal. Therefore there is no list of all the reals. — fishfry
I'll ask again:
But if you start from that there is no bijection, and then prove it by:
If there is a bijection then there is a surjection
There is no surjection.
Therefore, there is no bijection.
Isn't that a proof by contradiction?
— ssu — ssu
Only countably many interpretations of each sentence. — fishfry
a trip to the moon on gossamer wings — fishfry
We're talking about different things. I'm talking about formal theories and interpretations of their languages as discussed in mathematical logic, and such that theories are not interpretations. — TonesInDeepFreeze
Exactly. — TonesInDeepFreeze
I don't propound the notion that that approach could be adapted for natural languages too, but it doesn't seem unreasonable to me. — TonesInDeepFreeze
Seeing just that one phrase from the great song made my night. Such a soul satisfyingly beautiful song by a gigantically great composer. — TonesInDeepFreeze
enderton page ref please or st*u. second time i'm calling your bluff on references to your magic identity theory. — fishfry
It seems that from you I get extremely good answers. — ssu
Yes, Lawvere's fixed point theorem was exactly the kind of result that I was looking for. It's just typical that when the collories are discussed themselves, no mention of this. I'll then have to read what Lawvere has written about this. — ssu
And that not necessary is important for me. This is what TonesInDeepFreeze was pointing out to me also. I'll correct my wording on this. — ssu
I know you're kidding. But underneath there lies an actual point for me, which is that I don't think you know how insulting you are in certain threads when you read (if it can be called 'reading') roughshod over my posts, receiving them merely as impressions as to what I've said, so that you so often end up completely confusing what I've said and then projecting your own confusions onto me. — TonesInDeepFreeze
But I do appreciate that you quoted Cole Porter's so charming and magical lyric. And there was another special musical moment for me today, so my evening was graced. — TonesInDeepFreeze
From the OP at least I made the connection.I'm not sure how the subject came up. — fishfry
That's what really intrigues me. Especially when you look at how famous and still puzzling these proofs are...or the paradoxes. Just look at what is given as corollaries to Lawvere's fixed point theorem:It's interesting to know that all these diagonal type proofs can be abstracted to a common structure. They are all saying the same thing. — fishfry
Going back to the OP and the article given there, perhaps in the future it will be totally natural (or perhaps it is already) to start a foundation of mathematics or a introduction to mathematics -course with a Venn diagram that Yanofsky has page 4 has. Then give that 5 to 15 minutes of philosophical attention to it and then move to obvious section of mathematics, the computable and provable part. — ssu
IMO those concepts are far too subtle to be introduced the first day of foundations class. — fishfry
There's a lot that in mathematics is simply mentioned, perhaps a proof is given, and then the course moves forward. And yes, perhaps the more better course would be the "philosophy of mathematics" or the "introduction to the philosophy of mathematics". So I think this forum is actually a perfect spot for discussion about this.IMO those concepts are far too subtle to be introduced the first day of foundations class. Depending on the level of the class, I suppose. Let alone "Introduction to mathematics," which sounds like a class for liberal arts students to satisfy a science requirement without subjecting them to the traditional math or engineering curricula. — fishfry
It sure is interesting. And fitting to a forum like this. If you know good books that ponder the similarity or difference of the two, please tell.Truth versus provability is not a suitable topic near the beginning of anyone's math journey. IMO of course. — fishfry
First Grade in Finland the educational system then had this wonderful idea of starting to teach first grade math starting with ...set theory and sets — ssu
These are all mathematical truths, but they're not very interesting mathematical truths. — fishfry
In mathematics, truth is typically understood within the framework of logical consistency and proof. Here are a few key aspects of truth in mathematics:
Logical Consistency: Mathematical statements and propositions must be internally consistent. This means that there should be no contradictions within a mathematical system. For example, in Euclidean geometry, the parallel postulate is consistent with other axioms, but in non-Euclidean geometries, different parallel postulates lead to different but internally consistent geometries.
Verification through Proof: In mathematics, a statement is considered true if it has been proven using rigorous logical arguments based on accepted axioms and definitions. The process of proving involves demonstrating that the statement follows logically from these axioms and previously proven statements (lemmas).
Objective Reality: Mathematical truth is independent of human beliefs or opinions. Once a mathematical statement has been proven, it is universally accepted as true within the mathematical community. This aspect of objectivity distinguishes mathematical truth from truths in other domains, which may depend on subjective interpretation or observation.
Unambiguity: Mathematical statements are precise and unambiguous. Each term used in mathematics is defined rigorously, and the rules of inference and logical operations are well-defined. This clarity ensures that the truth of mathematical statements can be objectively assessed.
Scope of Truth: In mathematics, truths are often considered to be eternal and immutable once proven. For example, the Pythagorean theorem, once proven, remains true indefinitely and universally applicable within the domain of Euclidean geometry.
In essence, truth in mathematics is grounded in rigorous logical reasoning, proof, and adherence to accepted axioms and definitions. It is a fundamental concept that underpins the entire discipline, allowing mathematicians to build upon previously established truths to explore new areas and make further discoveries.
That needs work — TonesInDeepFreeze
There's a lot that in mathematics is simply mentioned, perhaps a proof is given, and then the course moves forward. And yes, perhaps the more better course would be the "philosophy of mathematics" or the "introduction to the philosophy of mathematics". So I think this forum is actually a perfect spot for discussion about this. — ssu
Of course it would be a natural start when starting to talk about mathematics, just as when I was on the First Grade in Finland the educational system then had this wonderful idea of starting to teach first grade math starting with ...set theory and sets. Ok, I then understood the pictures of sets, but imagine first graders trying to grasp injections, surjections and bijections as the first thing to learn about math. I remember showing my first math book to my grand father who was a math teacher and his response was "Oh, that's way too hard for children like you." Few years later they dropped this courageous attempt to modernize math teaching for kids and went backt to the "old school" way of starting with addition of small natural numbers with perhaps some drawings and references about a numbers being sets. (Yeah, simply learning by heart to add, subtract, multiply and divide by the natural numbers up to 10 is something that actually everybody needs to know.) — ssu
It sure is interesting. And fitting to a forum like this. If you know good books that ponder the similarity or difference of the two, please tell. — ssu
There's many things they don't teach in school when looking at what my children have to study. Usually the worst thing is when the writers of school books are too "ambitious" and want to bring in far more to the study than the necessities that ought to be understood.That sounds like the "New Math" they had when I was in school. I loved it but it was a failure in general.
I don't think they teach basic arithmetic anymore. It's a problem in fact. — fishfry
I looked at this. Too bad that William Lawvere passed away last year. Actually, there's a more understandable paper of this for those who aren't well informed about category theory. And it's a paper of the same author mentioned in the OP, Noson S. Yanofsky, from 2003 called A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points. Yanofsky has tried to make the paper to be as easy to read as possible and admits that when abstaining from category theory, there might be something missing. However it's a very interesting paper.Here's the general theorem in the setting of category theory. It's called Lawvere's fixed point theorem. Not necessary to understand it, just handy to know that all these diagonal-type arguments have a common abstract form. — fishfry
On a philosophical level, this generalized Cantor’s theorem says that as long
as the truth-values or properties of T are non-trivial, there is no way that a
set T of things can “talk about” or “describe” their own truthfulness or their
own properties. In other words, there must be a limitation in the way that T
deals with its own properties. The Liar paradox is the three thousand year-old
primary example that shows that natural languages should not talk about their
own truthfulness. Russell’s paradox shows that naive set theory is inherently
flawed because sets can talk about their own properties (membership.) Gödel’s
incompleteness results shows that arithmetic can not talk completely about
its own provability. Turing’s Halting problem shows that computers can not
completely deal with the property of whether a computer will halt or go into
an infinite loop. All these different examples are really saying the same thing:
there will be trouble when things deal with their own properties. It is with this
in mind that we try to make a single formalism that describes all these diverse
– yet similar – ideas.
The best part of this unified scheme is that it shows that there are really no
paradoxes. There are limitations. Paradoxes are ways of showing that if you
permit one to violate a limitation, then you will get an inconsistent systems.
Et tu? ChatGPT doesn't know anything about mathematical philosophy. It just statistically autocompletes strings it's been fed. — fishfry
Godel's completeness theorem, applied to group theory, says that any statement that's true for every group can be proved from the axioms of group theory. Similarly, there is more than one model of ZFC. The existence of various models of ZFC is analogous to the existence of different groups. Some statements are true in one model of ZFC and false in another. Such a statement is independent of ZFC.
But there are truths not derivable from the axioms. — TonesInDeepFreeze
https://www.hawking.org.uk/in-words/lectures/godel-and-the-end-of-physics
What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted.
I'm an antique. Truth for me is associated with proof. — jgill
In other words, whatever is provable is true. But it's not the case that whatever is true is provable. — TonesInDeepFreeze
People who believe that pure reason is the only source of meaning will never accept this, no matter how often you hammer it into their heads. — Tarskian
Any examples of those people come to mind? — Wayfarer
.https://g.co/kgs/3yTBNh7
a philosophical system that holds that every rationally justifiable assertion can be scientifically verified or is capable of logical or mathematical proof
(and that therefore rejects metaphysics and theism)
https://en.m.wikipedia.org/wiki/Hilbert%27s_program
Statement of Hilbert's program
...
Completeness: a proof that all true mathematical statements can be proved (in the formalism).
...
Gödel's incompleteness theorems, published in 1931, showed that Hilbert's program was unattainable for key areas of mathematics.
Positivists are like that. — Tarskian
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