Timeline for Reasoning Using Countable Subsets of Real Numbers
Current License: CC BY-SA 3.0
12 events
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| Feb 21, 2018 at 9:22 | comment | added | SSequence | @NikWeaver I have to say I find this very interesting (and hence the "extra" enthusiasm in comments), even if technically ill-equipped to understand any of it. In 1.5 you wrote: "I have also claimed that classical logic is appropriate only when quantifying over surveyable structures and that intuitionistic logic should be used when quantifying over indefinitely extendable but determinate concepts." This is a fairly remarkable observation. Though I do not quite agree with some remarks in 1.3 (discussion there is closer to my comfort zone because of being elementary). Thanks for the link. | |
| Feb 21, 2018 at 2:34 | comment | added | Nik Weaver | front.math.ucdavis.edu/0905.1675 | |
| Feb 21, 2018 at 1:47 | comment | added | SSequence | @NikWeaver When I used the words "set" and "countable" in the previous comments, I meant them in a strict platonic sense (for easier communication). Example, when a strict constructivist talks about a collection of "computable objects", it is countable in traditional sense. But to that person that it is hardly a "toy collection" (to many others with different positions, ofc it will be). I got a certain impression while glancing your comments in 1.4 of your paper in answer (but I definitely interpreted it wrong). Anyway, can you give a link to your work that you mentioned in above comment? | |
| Feb 21, 2018 at 0:58 | comment | added | Nik Weaver | @SSequence: I don't think this is the place for an extended discussion, but let me just say that you cannot both (1) work in a countable "toy" universe and (2) have the reals in your model coincide with the actual set of all real numbers. On my view the "set" of all reals is actually a proper class, and one can set up an axiomatic system for reasoning about them as such; I have worked on this, too. | |
| Feb 21, 2018 at 0:12 | comment | added | SSequence | represent the actual $\omega_1$ (even in our version of maths done with $A$). Thirdly, if we define the notion of $A$-collection for a set of $r$-numbers(similar to how I defined collection in the question), then clearly can't call $R$ isn't an $A$-collection. This means we would have to be careful and distinguishing between the sets $R$ (and possibly similar sets $B$ such as $B=R-C$, where $C$ is an $A$-collecion) and between "ordinary" $A$-collections. (COMPLETED) | |
| Feb 21, 2018 at 0:12 | comment | added | SSequence | That is, after suitable interpretation of $\mathbb{N} \rightarrow \mathbb{N}$ functions as $\mathbb{N} \rightarrow \{0,1\}$. There are two problems here. Now $\alpha$ is (in some sense) the analogue of $\omega_1$ in maths (involving our universe of functions $A$). We have again a very similar problem. If the choice of $A$ is based on a strong philosophical position, then $\alpha$ represents the "actual" $\omega_1$ (in a very genuine sense) in our version of mathematics done with $A$. But if one's choice of set $A$ is merely taken because of convenience then $\alpha$ clearly doesn't (cont.) | |
| Feb 21, 2018 at 0:09 | comment | added | SSequence | $r$-numbers by real numbers in all claims and be free in their reasoning). But if one's choice of set $A$ is merely taken because of convenience then we have some trouble, because we can't replace $r$-numbers by real numbers in our claims. In this case, we can still freely reason when talking "only" about $r$-numbers. But when we do try to reason about real numbers we will have to be quite careful (and probably very conservative) due to possibility that $\mathbb{R} \supset R$. Secondly clearly there must be some countable ordinal $\alpha$ whose well-order relation isn't contained in $A$ (cont. | |
| Feb 21, 2018 at 0:08 | comment | added | SSequence | I will follow the terminology in my question. Furthermore I will always use the term "set" in usual/traditional sense (to avoid too much confusion). Consider some countable set $A$ containing functions $f:\mathbb{N} \rightarrow \mathbb{N}$ and similarly consider the corresponding subset of real numbers $R$ (as described in question). Call the numbers in $R$ as $r$-numbers. There are at least three "problems" as I see. Firstly if due to some strong conviction/position one claims that the $r$-numbers actually do represent the reals precisely, then it's fine ..... one can replace (cont.) | |
| Feb 20, 2018 at 23:53 | comment | added | SSequence | @NikWeaver While I did not originally mention analysis (I almost included it), the answer is very much in the spirit of my question. Obviously I am not qualified to give any technical comment on this paper of yours (I did read the first few sections in basic detail). Also I read your other essay (for the general reader) introducing your program/philosophy (with some basic parts being more easily understandable than technical ones). I do have some, possibly very naive, observations (in the context of your paper in the answer). I will post them in comments to follow. | |
| Feb 19, 2018 at 20:17 | comment | added | Nik Weaver | @GerhardPaseman: I'm not sure what you mean by "a pertinent example". My whole paper is about how one can develop a version of core mathematics within $J_2$. Okay, Theorem 3.20 is a $J_2$ version of: a subset of $\mathbb{R}$ is compact iff it is closed and bounded iff it is bounded and closed under convergence of Cauchy sequences iff every sequence has a convergent subsequence. Is that what you're after? I mean the whole paper is a string of such results. | |
| Feb 19, 2018 at 19:05 | comment | added | Gerhard Paseman | Can you highlight/briefly describe a pertinent example? If not, words like "Example 3.2.1.1 on p 56 gives a motivating example B, which is like your A but contained in the complement" might help future readers as well as the original poster. Gerhard "Just Show Me The Spoon" Paseman, 2018.02.19. | |
| Feb 19, 2018 at 17:03 | history | answered | Nik Weaver | CC BY-SA 3.0 |