Timeline for Is the fact that ZFC implies that 1+1=2 an absolute truth?
Current License: CC BY-SA 4.0
30 events
| when toggle format | what | by | license | comment | |
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| Jan 20 at 6:06 | comment | added | user71091 | Self evident statement have as demonstration principle: what You see is what You get. Therefore, using self evident, we consider conclusion that follows: because it is like that, not because I want that. Because otherwise, we will end to define everything that make completely another Universe, different from real Universe, inaccessible for others, like often we may see lies or fairy tales. We cannot deduce them, because they are "invented irrationals", no deductive path to them. "1+1=2". | |
| Jul 23, 2022 at 14:04 | comment | added | gandalf61 | @BЈовић ZFC stands for a standard set of axioms for formalising mathematics called Zermelo–Fraenkel set theory with the addition of the axiom of choice (hence ZFC rather than just ZF). For more details see en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory and en.wikipedia.org/wiki/Axiom_of_choice. | |
| Jul 22, 2022 at 5:58 | comment | added | BЈовић | what means ZFC? | |
| Jul 21, 2022 at 22:28 | comment | added | mudri | It seems to have confused a load of people that you wrote “ZFC implies 1 + 1 = 2” (symbolically, ⊢ ZFC ⇒ 1 + 1 = 2), where you probably meant “ZFC proves 1 + 1 = 2” (symbolically, ZFC ⊢ 1 + 1 = 2). With the former, there is no connection between ZFC and the symbols “1”, “+”, “2”, and “=”; i.e, those symbols come from the theory determined by ⊢, rather than ZFC. With the latter, it's understood that the definitions are those we would make in ZFC (say, 1 is notation for {{}}, 2 is notation for {{}, {{}}}, &c). | |
| Jul 21, 2022 at 8:32 | answer | added | cmaster - reinstate monica | timeline score: 0 | |
| Jul 21, 2022 at 0:39 | comment | added | Robbie Goodwin | Can you rephrase that so the wording isn't open to question, please? For instance, what's a fact is a fact. What difference do you see between "fact" and "truth…" whether "absolute" or not? Are you suggesting that ZFC and 1+1=2 are mutually dependent, or not? If they are, how is that? If not, what does that say about your Question? | |
| Jul 20, 2022 at 20:47 | answer | added | Brian | timeline score: 0 | |
| Jul 20, 2022 at 19:21 | comment | added | TCooper | certainly not an answer to this question, but maybe what I wrote for this question, philosophy.stackexchange.com/questions/92058/…, is of interest to you? | |
| Jul 19, 2022 at 23:45 | answer | added | Davislor | timeline score: 0 | |
| Jul 19, 2022 at 22:58 | comment | added | Juan | @CharlesHudgins Yes! This is like Zeno's paradox applied to proofs (wonder if some similar solution could be found, like a convergent series of infinite premises). I guess this also explains the name of the article linked by gandalf61. | |
| Jul 19, 2022 at 22:52 | vote | accept | Juan | ||
| Jul 19, 2022 at 21:05 | comment | added | Charles Hudgins | Not exactly the same, but have you ever heard the parable of Achilles and the tortoise? | |
| Jul 19, 2022 at 17:31 | comment | added | Kevin Cathcart | And if you mix and match the peano definition of 1, with a complex number definition of 2, you can easily prove that 1 + 1 ≠ 2. In general mathematics, even asking is 1 ∈ 2 is true is a category error, because numbers are not really sets, but are merely modeled as such for proofs. | |
| Jul 19, 2022 at 17:31 | comment | added | Kevin Cathcart | A major limitation of trying to prove things back to ZFC is that for anything other than sets, you need to define the symbols, generally by modeling them as sets. But then you are effectively treating those definitions as additional postulates, and need to verify that they properly model the intended concept, and that the definitions are consistent with each other. For example if 1 and 2 are modeled Peano style, then 1 ∈ 2. But with say a complex number fomulation, that no longer holds.... | |
| Jul 19, 2022 at 13:48 | answer | added | Irishmanluke | timeline score: 1 | |
| Jul 19, 2022 at 11:45 | answer | added | AnoE | timeline score: 3 | |
| Jul 19, 2022 at 11:43 | answer | added | gnasher729 | timeline score: 0 | |
| Jul 19, 2022 at 11:41 | answer | added | Kristian Berry | timeline score: 0 | |
| Jul 19, 2022 at 11:04 | answer | added | gandalf61 | timeline score: 10 | |
| Jul 19, 2022 at 10:52 | comment | added | Criticizing Israel not allowed | You need definitions of 1, + and 2 to prove that 1+1=2. ZFC doesn't imply that 1+1=2, only that {{}} ∪ {{{}}}={{},{{}}} | |
| Jul 19, 2022 at 8:59 | history | became hot network question | |||
| Jul 19, 2022 at 5:37 | answer | added | David Gudeman | timeline score: 2 | |
| Jul 19, 2022 at 5:19 | answer | added | Nikos M. | timeline score: 6 | |
| Jul 19, 2022 at 4:08 | comment | added | Conifold | See also Leitgeb, On Formal and Informal Provability on Gödel's conception. | |
| Jul 19, 2022 at 4:01 | comment | added | Conifold | Gödel developed a technical notion of informal "absolute provability", as did Post, Tarski, Cohen, etc., see Leach-Krouse's thesis. But if you mean the philosophical sense of "absolute" then no, ZFC ⊢ 1+1=2 depends on human-all-too-human concepts, conventions, proof standards and practices, etc. All very non-absolute. | |
| Jul 19, 2022 at 3:00 | history | tweeted | twitter.com/StackPhilosophy/status/1549227639373692929 | ||
| Jul 19, 2022 at 2:53 | comment | added | Double Knot | Given Von Neumann universe cumulative-hierarchy interpretation as a class model of ZFC we immediately have V0=∅, V(n+1)=P(Vn) ⊢ V1={∅} ∧ V2=P(V1)={∅,{∅}}, then we interpret V1 as 1 and V2 as 2 thus 2=1+1 is well-defined in ZFC. Of course the same is commonly defined in PA using successor function S. So 2=1+1 is actually not a theorem of ZFC (or PA) but just a true analytic a priori proposition per logical positivists (or 2 is just a term well-defined in V). But due to vacuous truth your conditional could be absolutely true within FOL... | |
| Jul 19, 2022 at 1:38 | answer | added | philosodad | timeline score: 2 | |
| Jul 19, 2022 at 1:00 | comment | added | Kristian Berry | Saharon Shelah has spoken of "logical dreams" that would, I think, allow us to infer different arithmetic via forcing (I think), if the dreams could be realized somehow. Others think arithmetic represents an intuitive kernel of mathematical knowledge, and that we are more confident in intuitive arithmetic than any ZFC-like theory that seeks to obviate the appeal to intuition. Kant infamously thought that arithmetic was not logically necessary, hence not logically absolute. | |
| Jul 19, 2022 at 0:49 | history | asked | Juan | CC BY-SA 4.0 |