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July 23

Cantor's Diagonal Argument

(Disclaimer: I'm definitely not a mathematician.)

From what I understand, the eponymous article proves the set of all binary sequences is uncountably infinite by placing them in a grid, and then for the nth digit in each nth sequence inverting it to produce a sequence that could not have been included. E.g.
1 0 0 0 0 0 0 ...
0 1 1 0 0 0 0 ...
0 1 1 1 0 0 0 ...
0 1 0 1 0 0 0 ...
0 0 1 1 0 0 0 ...
0 0 0 0 0 0 0 ...
1 0 0 1 0 0 0 ...
...
gives 0 0 0 0 1 1 1 ... as its sequence. This then has a mapping created between it and the real numbers to prove the real numbers are uncountable, and so on.

What I've never quite understood, though, is---couldn't one put the natural numbers themselves (in binary) in such a grid, only extending leftwards into infinity instead of rightwards, and then perform the same process going the opposite direction? So e.g.
... 0 0 0 0 0 0 0
... 0 0 0 0 0 0 1
... 0 0 0 0 0 1 0
... 0 0 0 0 0 1 1
... 0 0 0 0 1 0 0
... 0 0 0 0 1 0 1
... 0 0 0 0 1 1 0
...
which gives a sequence ending in ... 1 1 1 1 1 1 1 (which will ultimately be an infinite number of 1s). Of course, that would then imply there is no mapping from the natural numbers to the natural numbers, which is absurd---so where is the error in that logic? Is it the fact that the grid extends leftwards instead of rightwards? (But if so, why then does it ruin things to view the numbers--which are still ultimately just infinite binary sequences, aren't they?--as some digits preceded by an infinite number of other digits, instead of some digits followed by an infinite number of other digits?) 2603:8001:4542:28FB:481A:8739:16B6:7E36 (talk) 06:35, 23 July 2023 (UTC)[reply]

The direction of the grid is irrelevant; we could reverse the endianness and agree that, for example, the decimal number 13 is represented by the binary sequence 1 0 1 1 0 0 0 ... . The issue here is that the binary representation of a natural number always contains a finite number of 1 digits. So your infinite sequence ... 1 1 1 1 1 1 1 fails to represent a natural. --Lambiam 08:29, 23 July 2023 (UTC)[reply]

So it is possible to have an infinite set of natural numbers, each one of which is finite? catslash (talk) 17:36, 23 July 2023 (UTC)[reply]

If by "finite" you mean "has a finite number of digits", then every natural number is finite, as @User:Lambiam noted. And yes of course it's possible to have an infinite set of natural numbers, such as the set of all even naturals, the set of all square naturals, and the set of all naturals itself. CodeTalker (talk) 20:03, 23 July 2023 (UTC)[reply]

One thing you might be confusing (a common confusion) is treating infinity as a number. Infinity isn't a number; it's just a mathematical concept of endlessness. Individual natural numbers cannot have an infinite number of digits to the left of the decimal place; they can only have an arbitrarily large, but finite number of such digits. Irrational numbers can have an infinite number of digits to the right of the decimal. This is because infinite decimals are convergent; which is to say that all terminating approximations of an infinite decimal become arbitrarily closer to the actual number in the Limit (mathematics). You can't write all of pi down, but you can get arbitrarily close to the actual value of pi, depending on how much you want to write. Infinite digits to the left of the decimal (excepting the degenerate case where the leftmost and all subsequent leftish digits are "zero") always diverge, which is to say you can't reach any specific real number by just terminating the series. The number 11111 and the number 111111 and the number 1111111 don't converge to a single real number, they are diverging and running off to infinity. Such a number is thus undefined. This is different from 3.14 and 3.142 and 3.1416 and such because each additional digit brings you closer to the same real number, it has a defined limit, being pi. --Jayron 32 16:34, 24 July 2023 (UTC)[reply]

Winning strategy of this game

What is the winning strategy of this game?

Using pokers spades 1 to 9, two players, first player can take a card from table, then second player takes another card from table, then the first player takes another card from table, etc., but a player can only have at most three cards on hand, he must put the first card to the table before he take the fourth card from table, and must put the second card to the table before he take the fifth card from table, etc., if the sum of the three numbers of a player’s three cards is 15, then this player wins. 42.76.54.75 (talk) 07:09, 23 July 2023 (UTC)[reply]

See Number Scrabble for an isomorphism with a game played on a board, the 3-by-3 tic-tac-toe grid. In the TTT version, each player has three pawns of their own colour. Initially none are on the board. The players, taking turns, either bring a pawn to the board, placing it on an empty square, or move own of their pawns on the board to an empty square. The player getting three pawns in a row wins. In classical TTT, the game can end in a draw when all squares are occupied without there being three in a row. In this game, that cannot happen: there are always at least three empty squares. It is not immediately clear, though, that any player can force a win; the game may go on forever. --Lambiam 08:42, 23 July 2023 (UTC)[reply]

No, when a player take the fourth card from table, he must put the first card to the table, he cannot choose to put the second or the third card to the table (also, a player can only have at most three cards on hand, a player must have these three cards sum to 15 to win). 113.196.184.161 (talk) 09:03, 23 July 2023 (UTC)[reply]

The only way to win in Number Scrabble is also with three cards. But my variant does not obey the first-in first-out requirement. --Lambiam 14:44, 23 July 2023 (UTC)[reply]

I guess from those rules one is allowed to pick up the same card as one has just discarded? NadVolum (talk) 11:25, 23 July 2023 (UTC)[reply]

No, not allowed to pick up the same card as one has just discarded. 118.170.4.163 (talk) 12:07, 23 July 2023 (UTC)[reply]

Pascal's triangle; does the nth row relate to n or to n-1??

The nth row of Pascal's triangle is related to n-1 in many ways. n-1 is the second and penultimate terms of the nth row for all n > 1. (The first row is simply 1, and it has no terms of that kind.) The terms also add up to 2^(n-1). The nth row is also used to classify the number of ways to pick a group of items out of n-1 total. (That is, the tenth row has the numbers of ways there are to pick out of a total of 9 items classified by how many items are being picked.) But it also has some relationships with n. There are a total of n terms. And the ratios of the terms are always equivalent to fractions whose numerator and denominator add up to n. Let's look at the tenth row (1-9-36-84-126-126-84-36-9-1) as an example. The ratios of the terms are the same as 1/9, 2/8, 3/7, 4/6, 5/5, 6/4, 7/3, 8/2, and 9/1 respectively. Which number, n-1 or n, are there more relationships with the nth row of Pascal's triangle?? Georgia guy (talk) 15:34, 23 July 2023 (UTC)[reply]

The best approach is not to use terms like

n

-th row, but to number the rows starting from 0, thus:

row 0 = $[1],$
row 1 = $[1,1],$
row 2 = $[1,2,1],$ ...,

which is the conventional enumeration as mentioned in our article Pascal's triangle. If you likewise number the items in a row starting from 0, row

~n

item

~k

is equal to the binomial coefficient

{n \choose k}

. --Lambiam 13:18, 24 July 2023 (UTC)[reply]

So why do children always learn to count with positive integers?? Do many mathematicians prefer to treat 0 as just another counting number for the same reason they prefer radians over degrees?? Georgia guy (talk) 13:20, 24 July 2023 (UTC)[reply]

To answer question 1) Zero (as a number, as distinct from "nothing" as a concept) is a comparatively new mathematical idea. 0#History has some background on the matter. Regarding your second question, I doubt anyone has taken a poll of every mathematician ever in existence, so I don't know how to provide you references for such a question. It should be noted that there are two different ideas you may be conflating; counting and indexing. No one (not even mathematicians) counts objects with zero representing a single object. The difference here is that you're not using the numbers to count, you're using them to label (or index) the rows. Zero is a fine number to start indexing at. You see it in other applications, like zeroth law of thermodynamics as well. --Jayron 32 13:49, 24 July 2023 (UTC)[reply]

Lots of Internet sites reveal that it has a reason. The first, second, and third laws were already named and well-established. The "zeroth law" was named because of its fundamental importance; too important for it to simply be called the fourth law. (Renumbering the other laws would be impractical.) Georgia guy (talk) 14:43, 24 July 2023 (UTC)[reply]

Indeed. --Jayron 32 15:07, 24 July 2023 (UTC)[reply]

There is something about this in the section Natural number § Modern definitions. Traditionally, the natural numbers go like 1, 2, 3, ..., but both in logic and in number theory it is more convenient to start at 0. Computer scientists tend to agree; read Edsger W. Dijkstra's "Why numbering should start at zero". --Lambiam 21:55, 24 July 2023 (UTC)[reply]