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
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.