Infinities and the Concept of Omnipotence

(This is an essay I wrote for school as a response for the first two chapters of Mystery of the Aleph by Georg Cantor. I ended up writing about omnipotence. I thought this was worth posting.)

Mystery of the Aleph: chapters aleph null and one

In the first chapter of The Mystery of the Aleph, it is described how Georg Cantor was working on the “continuum hypothesis” for the later years of his life. This hypothesis consisted of a single equation: 2^aleph null = aleph one. This problem, combined with a possible genetic disease, drove Cantor to insanity, and he was institutionalized many times.

The second chapter discusses Pythagoreans, and the first notions of infinity. The Pythagoreans discovered irrational numbers through the Pythagorean Theorem, and kept it a well-guarded secret. They practically went insane because of irrational numbers.

Later on, infinity was further explored and understood. The last sentence mentions that infinity was brought up in the medieval ages, in the form of religion. It is not elaborated, but it seems like this is referencing the supposed omnipotence of God. He is all-knowing and all-powerful, hence infinity.

This led me to think of a joke question: can God make a rock so big that he himself cannot lift it? I began to ponder the true meaning of this. What this really is is infinity minus infinity. The rock weighs X, and God’s strength is Y. So if X – Y is greater than one, He cannot lift it. And if X – Y is zero or less, he can lift it. So that raises the question, what is infinity minus infinity? A little bit of research turned up two possible answers: either it’s zero or it’s undefined. So that means either God can lift it, or we don’t know. But is that really true? If your strength is Y and an object weighs X and X = Y, wouldn’t that mean that you can lift it, but it takes an infinite amount of time? This is truly a difficult question that there is to straight answer to. Here’s the straight answer: it is not logically possible, much less physically so, to be all powerful.

Implications of An Infinitely Large Function if P ≠ NP

Prerequisite reading:
http://en.wikipedia.org/wiki/One-way_function
http://en.wikipedia.org/wiki/P_%3D_NP_problem

Let’s assume that one-way functions exist, meaning that there is something that’s unsolvable in reverse in polynomial time. So what if you’re using infinities? The polynomial of an infinity is still that infinity. But the exponential of an infinity is a greater infinity. So it may be possible to use a cipher that requires one to stay within the realm of one infinity. This would make it impossible to decrypt. Here’s an example.

P = plaintext
C = ciphertext

f(P) = C
where f is a function that takes aleph null steps. This works if there is some way to do aleph null steps, but it’s impossible to do aleph one steps. So the amount of time it takes to calculate is aleph null, but the number of steps it takes to solve is aleph one (x ^ aleph null = aleph one).

This all makes sense if you have infinite time. With infinite time you can do aleph null operations, but no more. Or if you can do infinite operations per second. But you can’t do aleph one operations in any amount of time, even infinite.

Powerprimes

WARNING: Math jargon ahead. If you don’t know what some of this means, you’ll have to do a little research.

A powerprime deals with the hyper-operation above multiplication: exponentiation. A powerprime is a number that cannot be expressed as a^b, just as a prime is a number that cannot be expressed as a*b. Here is a list of all the powerprimes up to 1000:

2 3 5 6 7 10 11 12 13 14 15 17 18 19 20 21 22 23 24 26 28 29 30 31 33 34 35 37 38 39 40 41 42 43 44 45 46 47 48 50 51 52 53 54 55 56 57 58 59 60 61 62 63 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 122 123 124 126 127 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 217 218 219 220 221 222 223 224 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 244 245 246 247 248 249 250 251 252 253 254 255 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999
proportion of powerprimes: 0.959 (This means that for this data set, 95.9% of the numbers are primes.)

As you can see, most numbers are powerprimes.

The following chart shows what percentage of numbers up to some point are powerprimes.

Up to…
10: 50.0%
20: 70.0%
50: 78.0%
100: 87.0%
200: 90.0%
500: 93.8%
1000: 95.9%
2000: 97.15%
5000: 98.2%
10000: 98.75%

Interestingly, unlike primes, it increases instead of decreasing. With primes, you can get a rough estimate of the number of primes up to n with the formula 1/ln(n). Is there any such formula for powerprimes? I was thinking that it might involve super-logarithms (tetration logarithms, see this). But to determine that I’d need a program that calculates super-logarithms, which, unfortunately, is not theoretically possible at this time. Nobody knows what would happen if you take the super-logarithm of a number that’s NOT a tetration of 2 numbers. Tetration remains very much a mystery. But I have a general idea of what a super-logarithm chart would look like, and I don’t think that the above chart is it. Here’s a super-log chart for what we do know. (Since we don’t know a value like e for super-logarithms, I’ll just use some other numbers.)

slog2 (2) = 1
slog2 (4) = 2
slog2 (16) = 3
slog2 (65536) = 4

So I will compare 1/slog2(n) to the proportion of powerprimes less than n.

N SLOG POWERPRIMES
2 1 0
4 0.5 0.5
16 0.33 0.6875
65536 0.25 0.995437622070312


They are not only opposite, but very different. The first three, when you add the slog result to the powerprime result, get a number that’s very nearly 1. But on the fourth, they aren’t even close to 1. Maybe if I look at the logarithm of n…

N LOG POWERPRIMES
2 1.44269504088896 0
4 0.434294481903252 0.5
16 0.360673760222241 0.6875
65536 0.0901684400555602 0.995437622070312

These come roughly close to adding up to 1. There could be a relationship there.

What about primes to powerprimes?

N PRIMES POWERPRIMES
2 0 0
4 0.5 0.5
16 0.375 0.6875
65536 0.099822998046875 0.995437622070312


These also seem to add up to one. Of course, that makes sense, because the proportion of primes is about the same as the logarithm formula, so they should have about the same result.

So primes and powerprimes come close to adding up to one. Coincidence? Possibly. But what is the relationship? That is the question.

Pseudorandom Number generator

I have an algorithm for a pseudorandom number generator which I’m pretty sure is random for practical purposes.

Here is some concept code for the algorithm.

n = 10*
a = 2^50*

100* times: **

       100* times: ***
            b = 2*a
            c = the last n digits of b

       if c > a
            return 1
       else if c < a
            return 0

*this number is somewhat arbitrary, and can be changed to another number that is close to this number
**It will return this many numbers.
***It has to skip some numbers, because otherwise it will never return a 1 or 0 more than four times in a row.

So this will randomly return 1s and 0s.

Here’s what my random number generator returns with the above arbitrary numbers:
0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 1 1 0 1 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 1 1 0 1 0 0 0 0 1 0 1 0 1 0 0 0 1 0 1 1 1 1 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1
59 zeros, 41 ones

Here’s what my computer’s random number generator returns:
1 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0 0 0 1 0 1 1 1 1 1 1 1 1 1 0 0 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 0 0 1 1 1 1 0 0 1 0 1 0 1 1 0 0 0 1 0 1 1 0 0 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 1 0 1 0 1 0
43 zeros, 57 ones

They are very similar, in a general sense. The both have about the same number of long sequences (>5 in a row), medium sequences (3 or 4), and short sequences (1 or 2).

Can you identify the pattern here?
0 1 1 0 0 0 0 1 1 0 0 0 1 1 0 0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 0 0 1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 1 1 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 1 0 1 1 0 1 0 0 1 1 1 0 0 1 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 1 1 1 1 0 1 1 0 1 0 1 0 0 0 1 1 0 0 0 1 1 1 1 0 1 0 1 0 0 0 1 1 0 1 1 0 0 1 0 1 0 0 1 1 1 1 0 0 1 0 0 1 1 1 1 0 0 0 1 0 1 0 1 0 1 0 0 1 1 1 0 1 1 1 0 0 0 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 0 0 1 1 1 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 1 0 1 1 1 0 0 1 0 1 1 0 1 1 1 1 0 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 1 0 1 0 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 1 1 1 1 1 0 1 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 1 1 1 1 0 1 0 0 1 0 1 1 0 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 1 0 1 1 1 0 0 1 1 1 0 0 1 0 1 1 0 1 1 0 1 0 1 1 1 0 1 1 1 0 0 0 0 1 0 1 1 0 1 0 1 0 1 1 0 0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 1 1 0 0 1 0 0 0 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 1 1 1 0 0 1 0 0 0 1 1 0 1 1 1 1 1 0 1 0 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 1 1 1 1 1 1 0 0 0 1 1 1 1 0 1 0 1 0 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 0 0 1 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 1 1 1 0 1 1 1 0 0 0 1 1 1 0 0 1 1 0 1 1 0 1 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 0 1 1 1 0 0 1 0 1 1 0 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 0 0 0 0 0 1 0 1 0