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"you claim that when he said he would "pick a black female" for SCJ, that wasn't really true, and all sorts of other criteria were considered." I never said that "pick a black female" wasn't really true; my position was that this was along with potential candidates' other qualifications already being known. By claiming that this isn't true, what we're left with is the conclusion that black + female were the sole criteria. Where is the strawman? I don't think anyone is contending this This is a false dilemma, and at the crux of the issue here. There is nothing about "I will pick a black woman" that says other factors weren't considered. It is an assumption, and one that nobody here has backed up with evidence. (edit to add: IOW, you are essentially claiming that you know that Biden did not do anything to vet people before his announcement and there is no way for you to know this, so how can you possibly insist that it is true? "I will do X" is not the same as "I will do X and only X") I'm waiting for J.C. MacSwell to come by and admonish you for your strawman

I obviously can't know the thoughts of Biden so I'll only speak for myself, but I suspect based on past actions and comments that Biden feels roughly the same way. The selection of KBJ was racist in neither intent nor fact, even though the selection criteria included being a black woman. The selection included the anticipated benefits of gaining Biden popularity with whites, asians, rich people, poor people, and of course blacks among others. The primary benefit was to finally recognize the equality of a long marginalized group while placing an outstanding liberal judge on the Supreme Court. Since Biden is unable to control the perceptions of others, and since any methodology of selecting a black woman was likely to cause turmoil anyway, he simply chose the method he felt would work best and went for it, and let the chips fall where they may. His goal was NOT to minimize the discomfort his selection would have on certain people. When it is a question of doing the right thing, I believe you do it and simply accept that people who don't like it will find a reason to object. The objections to Biden's method are a natural part of any significant public action and go a long way to help pull people into the future; talking about previously unheard of changes makes them more common and easier to accept. The concern about how Biden went about this is reminiscent of the reaction to blacks sitting at the white lunch counter, interracial marriage, gay marriage, transgender students in schools sports, women working "men's" jobs, and a long list of other changes. People don't like change and prefer to be eased into it slowly. The first time these things happened the naysayers were up in arms, complaining about how they are going about it, but at this point the uproar would come if you tried to STOP a black person from sitting at a lunch counter. Personally I'm a fan of ripping the bandage off all at once. It may hurt, and it may even not work on the first try, but it is the right thing to do. We shouldn't deprive people of what is rightfully theirs due to the discomfort of others.

The difference between competitive sports and politics is that in competitive sports there’s a clear, objective measure of “better”. Tiger Woods has won because he did better on the golf course. There is no single objective “better” when it comes to the judiciary. There is no “best judge” that can be objectively determined, merely a pool of judges with better qualifications, among whom the President picks. So by picking a black woman from that pool, the President isn’t deliberately choosing a worse candidate - merely a particular one from among a pool of similarly qualified candidates.

So it is a set that can be indexed by ordered pairs of integers, namely the base and the prime whose square root is the exponent. It will again be countable, as the set of ordered pairs of natural numbers has a bijection with the set of natural numbers. You keep trying to use the fact that these numbers are transcendental as if that gives you a natural connection to uncountability. It doesn’t. The fact that the set of all transcendentals is large doesn’t mean that any particular subset will be large.

What do you mean by {ℕi'}? If you mean the set of all i’-indexed sequences of natural numbers, that set has cardinality equal to that of the continuum. It clearly has the same cardinality as ℕℕ, and it’s not hard to establish that that has the same cardinality as the reals.

The set of all transcendentals has cardinality equal to that of the reals. The set of all square roots of primes has cardinality equal to that of the natural numbers, by rather obvious bijection. Neither of those have “intermediate” cardinality. If you are claiming to have constructed a set of intermediate cardinality, what is it? And your point is? Cardinality is not about underlying members. Just because the elements of a set are transcendentals and there are lots of transcendentals doesn’t mean that set is large. The cardinality of the set containing just pi is still 1.

In that case, why not give people the supplement instead, bypassing the pig and chicken slaughter? What for? If bacteria can do it fast and cheaply on a large enough scale, why implement a more expensive and complicated procedure?

First and foremost, KJB was. Obviously not by getting chosen, but by her race and gender being used for political purposes. But I guess that can be swept under the rug by anyone that feels she benefited from the overall process.

There's a test called Winograd schema, named after Terry Winograd, an early computer scientist. https://en.wikipedia.org/wiki/Winograd_schema_challenge Winograd's original example was: The city councilmen refused the demonstrators a permit because they [feared/advocated] violence. A human knows that "they" refers to the councilmen if the verb is feared, and to the demonstrators if the verb is advocated. An AI presumably would have a hard time with that, because you need to know what the words mean. Winograd worked in the pioneering days of AI, I have no idea if the current generation of machine learning algorithms can figure these kinds of things out. Although the following page refers to a Winograd schema challenge from 2016, and there's a link on that page to a report on Winograd schemas issued in 2020. So evidently these are still considered an interesting challenge for AIs. This page has a lot of other relevant links. https://cs.nyu.edu/~davise/papers/WinogradSchemas/WS.html Computers have no ideas of anything, they only consist of switches that can each be in one of two states, 1 or 0. The meaning is supplied by the humans who program and interpret the bit patterns. Just as we can program a computer to do finite arithmetic, we can program a computer to do infinite arithmetic. For example given two cardinal numbers [math]\aleph_\alpha[/math] and [math]\aleph_\beta[/math], we have [math]\aleph_\alpha + \aleph_\beta = \aleph_\alpha \cdot \aleph_\beta = \max(\aleph_\alpha, \aleph_\beta)[/math] In other words cardinal addition and multiplication are ridiculously simple; the sum and product of two cardinals is just the larger of the two. And cardinal subtraction and division aren't defined. Computers can also represent many of the countable ordinals, namely all the ordinals below the Church-Kleene ordinal, which is the first non-computable ordinal. The computer wouldn't have any idea that it's computing with transfinite numbers, any more than it knows that it's adding 1 + 1. It's just flipping bits according to instructions. Computers don't have ideas. That's critical to understand in all discussions of machine intelligence. Computers flip bits. Even the latest deep neural net is just flipping bits. You can see that by remembering that the AI programs run on conventional hardware. Their code is ultimately reducible to bit flipping; and, as Turing pointed out, to making marks with a read/write head on a long paper tape. You could in principle execute AI code using pencil and paper. That doesn't mean a computer can't be conscious or self-aware. Some people (not me) believe that the human brain operates by bit flipping. I think that's wrong, but some people do believe it. But whatever it is that AI's do, they do not do anything different than what regular vanilla computer programs do. They're organized cleverly for data mining, but they are not doing anything computationally new.

It’s trivial to construct a bijection between the set of prime numbers and the set of square roots of prime numbers. The set of prime numbers is obviously countable, so the set of square roots of prime numbers is countable, too. It sounds like you are trying to talk about the set of digits of the square roots of prime numbers. This is still countable, though slightly less obviously, but it is not the same as the set of square roots of prime numbers. It should be, if you can make clear statements. My point is that you have not done so. I have been: I can’t determine what statements you are proposing to be true.

A better article would be able to articulate that axiom choice is much less about what’s true and more about what’s convenient. These mathematicians have come up with some axioms which work together in an apparently surprising way, and therefore may be more convenient. That doesn’t make them more true, merely more likely to be used in the future.

There is something in this thread here which reminds me of MLK's letter from Birmingham jail:

You’ve returned to word salad; I can’t determine even what statements you are proposing to be true beyond the supposed conclusion. The pop-sci article is…poorly informed. This statement: makes no sense - one axiom implying another is not a way to gauge whether they are “more likely”. And the first half of the article keeps implying there’s a “battle” over whether the continuum hypothesis is “true”, before the second half reveals the actual state of affairs: it is independent of our current axiom set, meaning it could freely be true or false, depending on which model you choose.

You are going to be a lot more precise to make any headway. What is "the Natural/Real set" ? And what properties do the members of 'the set of square roots of prime numbers enjoy' that have already been proven ?

I should alsao like to point out that what Mathematicians mean by the word 'Field', particularly in relation to number theory, and what Physicists mean are quite different. You should disentangle the Physics and the Mathematics please.

Except that INow has been throwing around the "butt-hurt" comment far longer than I have. To say nothing of the insinuations of racism, from you. Exactly. And "prejudice' Means to judge beforehand, since you like definitions. As in J Biden 'judging beforehand' That his pick would be a black woman. Is there anything else I need to explain to you ? Or am I just being vitriolic ? I'm sure he would have, or at least come by to tell me to settle down because it's not that big an issue ( he's a lot more sensible than I ), but he lives too far away.

Unlike you I don't need to consult dictionaries to know what a word means, or who's full of shit. Any way you slice it, it means a bias, or preconceived notion, idea, or opinion about someone/thing. The active word, in this particular context, being preconceived, as in preannouncing your selection before actually making it. No I admitted to being less sensible than JC. Why don't you look up 'sensible' in the dictionary. @MigLignoring this are we? Not at all. I see no difference. Both are filling a vacancy, and preannouncing your selection is a preconceived notion about who you will rent to, or, who you will nominate to the SC. He did do the right thing. Did he 'improve things by the pre-annoucement about 'picking a black woman', or did he make things slightly worse ? That is what i have a dim view of; the pre-announcement. He eventually picked the candidate he wanted ( and we need ), but the pre-announcement of the selection criteria, involving skin color and gender, did not improve things but made things slightly worse and gave people the excuse to gripe about the selection process. I think the pre-announcement did not need to be made. If anyone can point out any benefit arising from it, I will stop getting MSC all worked up.