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Mirrors > Home > ILE Home > Th. List > 1kp2ke3k | Unicode version |
Description: Example for df-dec 9344, 1000 + 2000 = 3000.
This proof disproves (by counterexample) the assertion of Hao Wang, who stated, "There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula would be forbiddingly long... even if (one) knows the definitions and is asked to simplify the long formula according to them, chances are he will make errors and arrive at some incorrect result." (Hao Wang, "Theory and practice in mathematics" , In Thomas Tymoczko, editor, New Directions in the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc., Boston, 1986. (QA8.6.N48). The quote itself is on page 140.) This is noted in Metamath: A Computer Language for Pure Mathematics by Norman Megill (2007) section 1.1.3. Megill then states, "A number of writers have conveyed the impression that the kind of absolute rigor provided by Metamath is an impossible dream, suggesting that a complete, formal verification of a typical theorem would take millions of steps in untold volumes of books... These writers assume, however, that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms. This is not necessary. There is no reason not to make use of previously proved theorems rather than proving them over and over... A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used. Of course, this is how ordinary informal mathematics is normally done anyway, but with Metamath it can be done with absolute rigor and precision." The proof here starts with , commutes it, and repeatedly multiplies both sides by ten. This is certainly longer than traditional mathematical proofs, e.g., there are a number of steps explicitly shown here to show that we're allowed to do operations such as multiplication. However, while longer, the proof is clearly a manageable size - even though every step is rigorously derived all the way back to the primitive notions of set theory and logic. And while there's a risk of making errors, the many independent verifiers make it much less likely that an incorrect result will be accepted. This proof heavily relies on the decimal constructor df-dec 9344 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits. (Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.) |
Ref | Expression |
---|---|
1kp2ke3k | ;;; ;;; ;;; |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn0 9151 | . . . 4 | |
2 | 0nn0 9150 | . . . 4 | |
3 | 1, 2 | deccl 9357 | . . 3 ; |
4 | 3, 2 | deccl 9357 | . 2 ;; |
5 | 2nn0 9152 | . . . 4 | |
6 | 5, 2 | deccl 9357 | . . 3 ; |
7 | 6, 2 | deccl 9357 | . 2 ;; |
8 | eqid 2170 | . 2 ;;; ;;; | |
9 | eqid 2170 | . 2 ;;; ;;; | |
10 | eqid 2170 | . . 3 ;; ;; | |
11 | eqid 2170 | . . 3 ;; ;; | |
12 | eqid 2170 | . . . 4 ; ; | |
13 | eqid 2170 | . . . 4 ; ; | |
14 | 1p2e3 9012 | . . . 4 | |
15 | 00id 8060 | . . . 4 | |
16 | 1, 2, 5, 2, 12, 13, 14, 15 | decadd 9396 | . . 3 ; ; ; |
17 | 3, 2, 6, 2, 10, 11, 16, 15 | decadd 9396 | . 2 ;; ;; ;; |
18 | 4, 2, 7, 2, 8, 9, 17, 15 | decadd 9396 | 1 ;;; ;;; ;;; |
Colors of variables: wff set class |
Syntax hints: wceq 1348 (class class class)co 5853 cc0 7774 c1 7775 caddc 7777 c2 8929 c3 8930 ;cdc 9343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-sub 8092 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-5 8940 df-6 8941 df-7 8942 df-8 8943 df-9 8944 df-n0 9136 df-dec 9344 |
This theorem is referenced by: (None) |
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