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I have already said that for a finite set, it is obvious that the number of subsets is greater than the number of elements, and that it doesn't require the diagonal argument to prove this. The reason I did apply the diagonal argument was to demonstrate the procedure of the diagonal argument, rather than to actually prove anything. But Cantor's theorem is not just about finite sets. Although Cantor's theorem is true for finite sets, including the empty set, it is for infinite sets that the theorem is intended to be applied. And it is for infinite sets that the diagonal argument demonstrates its power. No. Although it is true for finite sets that a proper subset has fewer elements than the set, it is not true for infinite sets. Indeed, it is a defining property of infinite sets that they can be mapped one-to-one onto proper subsets. As far back as Galileo Galilei, it was demonstrated that the number of square numbers is equal to the number of numbers themselves: [math]1 \longleftrightarrow 1[/math] [math]2 \longleftrightarrow 4[/math] [math]3 \longleftrightarrow 9[/math] [math]4 \longleftrightarrow 16[/math] [math]5 \longleftrightarrow 25[/math] [math]6 \longleftrightarrow 36[/math] [math]7 \longleftrightarrow 49[/math] [math]8 \longleftrightarrow 64[/math] [math]...[/math] [math]n \longleftrightarrow n^2[/math] [math]...[/math] Indeed, one can even form a one-to-one correspondence between [math]n[/math] and [math]2^n[/math] for all [math]n \in \textstyle \mathbb {N}[/math]: [math]1 \longleftrightarrow 2[/math] [math]2 \longleftrightarrow 4[/math] [math]3 \longleftrightarrow 8[/math] [math]4 \longleftrightarrow 16[/math] [math]5 \longleftrightarrow 32[/math] [math]6 \longleftrightarrow 64[/math] [math]7 \longleftrightarrow 128[/math] [math]8 \longleftrightarrow 256[/math] [math]...[/math] [math]n \longleftrightarrow 2^n[/math] [math]...[/math] This allows one to create the following one-to-one correspondence of the natural numbers to subsets of the natural numbers: [math]1 \longleftrightarrow 00000000...[/math] [math]2 \longleftrightarrow 10000000...[/math] [math]3 \longleftrightarrow 01000000...[/math] [math]4 \longleftrightarrow 11000000...[/math] [math]5 \longleftrightarrow 00100000...[/math] [math]6 \longleftrightarrow 10100000...[/math] [math]7 \longleftrightarrow 01100000...[/math] [math]8 \longleftrightarrow 11100000...[/math] [math]...[/math] Note that applying Cantor's diagonal argument to this gives [math]11111111...[/math] as the representation of the subset of the natural numbers that is not in this list. That is, it appears as if the set of the natural numbers itself is the "final" subset of the set of all subsets. But note that the above list is not a list of the set of all subsets but is a list of the set of all finite subsets, and this can be mapped one-to-one onto the set of natural numbers. The subset corresponding to [math]11111111...[/math], the set of natural numbers itself, is not a finite subset so it doesn't belong to the above list of subsets. In other words, Cantor's diagonal argument is only excluding a subset that would be excluded anyway.

You are probably correct.

I suspect the reason that blame keeps getting heaped on Israel more than Hamas is because Hamas is no longer rampaging through Israel, but Israel is still rampaging through Gaza. Every time someone kills a child they invite criticism. In the beginning of this most recent mess Hamas received the lion's share of rebuke. Now that Israel is on the offensive it is they who receive the lion's share of the rebuke. I personally don't find that surprising at all. Once the fighting dies down I suspect there will be a more even-keeled evaluation of who is to blame for what.

This was the debate that I saw the video recording of: https://www.zdnet.com/article/ai-debate-3-everything-you-need-to-know-about-artificial-general-intelligence/ Gary Marcus moderated it (he's on the same level as LeCun but a much more likeable guy), Chomsky was one of the participants. Marcus did a vid for Wired recently, geared towards the layman: https://www.wired.com/video/watch/tech-support-ai-expert-answers-ai-questions-from-twitter Awesome! I'm again up to 30 red, now I have my badge of honor back!

I don't know whom you are referring to but it could be taken to be insulting. Nothing to do with me, but I see you are accumulating negative rep points, be warned about the story of the dog. There is actually quite a bit of truth in the rest of your post, pity there were so amny points each too short and without support. Nevertheless Yes Murray did the work on the Zebras, as was acknowledged in my reference. I assume by Dr Falsi you mean Dr Fauci ? I also assume you think isolation was wrong ? I would be interested to see you apply the very simple Laplace equation to obtain the kind of chemical and biochemical phenomena we are talking about. What function are you connecting in space and time ? You are correct in that there is a substantial probability element in epidemic modelling. This link was proven and started more than a century before Covid By Ross.

Yep. Good essay. And comments from you. There is no morally tenable position that involves victimized groups claiming a basis to victimize another group. One would hope that functional adults could at some point own up to the Nakba and ensuing ghettoization. The process where people in a nation develop some historical self-awareness can happen, though it's often slow. In the US, for example, a huge majority now understand the American Nakba of indigenous people, and that the European settlers were culpable in acts of genocide and dispossession. At some point, we were able to shift from viewing Native Americans solely as terrorists to viewing them as brutally murdered and displaced peoples trying to keep some shred of their former lives, lands, and cultures. As with so many such situations, land is key. And displacement often is run on the principle that my people can make better use of that land than you do, we are better, more civilized, and some ancient text proves it!

That goes towards the main thrust of my argument. The first point is that using proximate events (i.e. attacks) will result innocent life lost (on either side, and we can ignore the relative scale to avoid complications). The second is that the status quo inevitably will lead to attacks. From this it follows that the current situation is a morally untenable one, as it require the acceptance of regular innocent deaths. The point I was therefore trying to make is that anything other than a large-scale redesign of the relationship is necessary, and so far a two-state solution as the endpoint has been the only theoretically viable option. This also means that groups actively eroding this or other peaceful paths, are culpable in the ensuing death cycle. So yes, Hamas clearly are the proximate perpetrators, but the existing system is the overarching framework resulting in them (or eventually other groups) to attack Israeli civilians. And as it stands, actors on both sides have been torpedoing peace efforts, with civilians bearing the outcomes. That is not to say that there are any clear paths ahead, but I do think that it is necessary to focus and elevate voices that work toward that goal, rather than allowing populists and terrorists to frame the condition and direct the outcome. Building peace is so much harder than waging war and requires a disproportionate effort. But it is the only way out of it. In other words, the gist of the argument is that we should not focus on proximate conflicts as those will obscure any paths out of violence. Somewhat unrelated to this point, but touching on many aspects presented in this thread, there is an excellent assay on the New Yorker by Gessen that is worth a read. It starts with a Jewish perspective on anti-Semitism but also links that to the situation in Gaza. It is well worth the time to read: https://www.newyorker.com/news/the-weekend-essay/in-the-shadow-of-the-holocaust

! Moderator Note OK, 17 pages into this discussion, and I'd like to know if any of the input you've gotten has persuaded you to soften your position, or if it's been of no value and you stand by it adamantly. Nobody wants to discuss any subject with a preacher, someone who has no intention of being persuaded by any argument. Please give a brief summary of what, if anything, you've taken on board wrt this discussion.

What “phenomenon” are you talking about, and in what way do you think invariance of scale applies to it?

Excellent summary. +1

That's nowhere near the standard of silly mistakes I make. +1

I have a habit of re-reading my comments and I often cringe at minor mistakes I make in spelling, punctuation, and grammar. I will leave minor errors in place when they do not significantly distract from my conveyance of meaning; however, caparative is not the word I intended here. In the above paragraph, the word should have been comparative as I meant a process of evaluating similar and dissimilar qualities. My apologies for the confusion.

I think the Hamas part is very much implied, but perhaps too implied. That being said, the rest mostly referred to the otherwise hard to understand situation why Hamas has come to power in the first place. I think it is pretty obvious that Hamas has no intention to make lives better for Gazans (though potentially at the beginning some might have thought that). As mentioned, the issue is the ongoing campaign, and basically as some analysts and also Barak have mentioned, the issue is that the the suffering heaped on civilians (not Hamas) is depleting the goodwill that Israel had from suffering the violent attack by Hamas. They were therefore saying that Israel needs a targeted and timed action, but so far it is no unclear where things are going to end up. It does not help that some officials have floated the permanent displacement of Gazans and I think the original 24h deadline for Gazans to leave the North Gaza, without any assurance of a safe path (that came later, but the damage was done). That is a fair assessment . I think the issue I had with some justifications is the lack of a scope (I have mentioned commando-style actions as suggested by the US for example). But lack of announcements in that regard and again and in some cases pushing the likelihood of permanent displacement (with others contradicting those statements) does not really inspire confidence. As I said, I have simple thoughts and the thoughts said that at least we should spare the children (and then perhaps move up the morality ladder a bit if we hadn't failed that step already. And again, it does not matter whose child it was). I know you are trying to find policies that contradict this particular moral stance but as we have discussed in threads on abortion, it depends on the developmental stage of the fetus and its likelihood of survival, but also the risk to the mother. And in this context, late term abortion is a conservative talking point that misses its mark entirely. Medically, late term refers to pregnancies past 41 week gestation. I.e. if birth happens later than expected (40 weeks). Of course there is no abortion in that time frame. Moreover, fewer than 1% of abortions happen after the 21st week and I am not sure whether any of those are without some medical indication. But I suspect that it was more a jab at my simple moralism rather than a serious argument.

Label the States Even with a blank map, a lot of people can only name 45-50 of the 64 states.

On the contrary, it means that in war , mistakes and accidents do happen. The 'fog' and all that ... But, if you don't have an open mind, and have already convinced yourself of someone's guilt, it is easy to assume the worst about them.

Co-write the script with your son, get Sandra Bullock and George Clooney to star, and shoot the space scenes in a studio?

200 lbs, falling 12 ft, and you catch yourself with your fingertips. 85 times ! you have no concept of reality, or you watch too many movies.

Triggering volcanoes to combat global warming ... That's like cutting off an arm in an effort to lose weight.

What if the sky was made of concrete? Forget it. We can't hope to control volcanic eruptions.

Over-confidence is antithetical to a good scientific approach.

If it's a detonator, it only needs to work once. So it's a slightly different scenario. But, since it's a detonator, I'm not going to help.

My usage of the term is clear enough. You don't get to redefine it. Asking loaded questions ('have you stopped beating your wife?' being an infamous example) is not a mark of wisdom. It is a mark of petty spitefulness. As is having a dig at my age.

He’s been soapboxxing for a long time now. Chip on shoulder. Axe to grind. Seems to have an agenda…

These guys are a little late to the party. The Universe runs according to probablities and The Minimum Principle. Darwin's theory is really just a specific case as it relates to biology. Self-organization is a natural consequence of those two rules. S Kauffman's The Origins of Order is an excellent summary of this, and JD Murray has done a ton of work in many different areas of biology applying it. The zebra pictures above are from his work and summarized in his book Mathematical Biology. If Dr. Falsi had read that, he wouldnt have told us to isolate to deal with CoViD....BTW- the stripe pattern of Zebras is a special application of the Laplacian equation that also applies to epidemics-- the zebra's stripes and the wave pattern of an epidemic are mathematically related.

There has been one correction to my article published on Towards Data Science as a result of my activity on this board (I tried getting Mindmatters to make the change to their republication but they didn't): Original sentence: Corrected sentence: As I have previously stated, that doesn't change the point because models never reflect the actual. This has been a common observation, as seen through various related aphorisms: Correlation does not imply causation https://en.wikipedia.org/wiki/Correlation_does_not_imply_causation All models are wrong, some are useful https://stats.stackexchange.com/questions/57407/what-is-the-meaning-of-all-models-are-wrong-but-some-are-useful The map is not the territory https://en.wikipedia.org/wiki/Alfred_Korzybski Also, as I have previously stated in the thread, at least 50% of my article came from discussions on public forums, namely the entire section "Responses to counterarguments" This contrasts to utterly useless quips from certain users like iNow and StringJunky above. They are evidently only interested in the "meta." (e.g. "You are wasting your time," "He has chip on his shoulder," "He is overconfident," and other junk remarks) Please remind them to focus on the points, not on some pseudopsychoanalytic prods on perceived modus operandi. Sure, people on Reddit did that quite a bit, but at least even they coughed up useful items to add (re: exhaustive modeling, answering to accusations of intuition pumping from Dennett fanboys, etc), in this sense, iNow/StringJunky are being less than useless, worse than Reddit. Heck, even the idiot moderator on Ars(e) Technica gave me the lone "gift" of special pleading accusation before he banned me without allowing me to even reply once (because he thought that was the last word or something) Did I mention that he even deleted my post (yes, the article I wrote) after banning me? p.s. if they even bothered reading the thread, they'd already SEEN what exactly the "agenda" is. I've already mentioned the 2021 UN agency ban on AI legal personhood multiple times Bad analogy when it comes to machine consciousness since anyone who has digested the points of the argument would see that intentionality has nothing to do with any kind of machine performance. Well, iNow sure didn't read it- Doubt anyone else besides us would read it. Believe me, loads of people on my LinkedIn are also saying that the LLM approach to AI is a dead end. Chomsky mentioned GOFAI and guess what? There are talks of going back to it. Not entirely, but incorporation. That's one way to get any semblance of grounding. I saw a recording of a debate/discussion that occurred at the start of this year, and Chomsky said similar things as far as I could remember. Let me see if I can go find a link to it. It's chock full of today's authorities on AI.