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Mirrors > Home > ILE Home > Th. List > 1kp2ke3k | Unicode version |
Description: Example for df-dec 9297, 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 9297 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 9107 | . . . 4 | |
2 | 0nn0 9106 | . . . 4 | |
3 | 1, 2 | deccl 9310 | . . 3 ; |
4 | 3, 2 | deccl 9310 | . 2 ;; |
5 | 2nn0 9108 | . . . 4 | |
6 | 5, 2 | deccl 9310 | . . 3 ; |
7 | 6, 2 | deccl 9310 | . 2 ;; |
8 | eqid 2157 | . 2 ;;; ;;; | |
9 | eqid 2157 | . 2 ;;; ;;; | |
10 | eqid 2157 | . . 3 ;; ;; | |
11 | eqid 2157 | . . 3 ;; ;; | |
12 | eqid 2157 | . . . 4 ; ; | |
13 | eqid 2157 | . . . 4 ; ; | |
14 | 1p2e3 8968 | . . . 4 | |
15 | 00id 8017 | . . . 4 | |
16 | 1, 2, 5, 2, 12, 13, 14, 15 | decadd 9349 | . . 3 ; ; ; |
17 | 3, 2, 6, 2, 10, 11, 16, 15 | decadd 9349 | . 2 ;; ;; ;; |
18 | 4, 2, 7, 2, 8, 9, 17, 15 | decadd 9349 | 1 ;;; ;;; ;;; |
Colors of variables: wff set class |
Syntax hints: wceq 1335 (class class class)co 5825 cc0 7733 c1 7734 caddc 7736 c2 8885 c3 8886 ;cdc 9296 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 ax-setind 4497 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-addcom 7833 ax-mulcom 7834 ax-addass 7835 ax-mulass 7836 ax-distr 7837 ax-i2m1 7838 ax-1rid 7840 ax-0id 7841 ax-rnegex 7842 ax-cnre 7844 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-br 3967 df-opab 4027 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-iota 5136 df-fun 5173 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-sub 8049 df-inn 8835 df-2 8893 df-3 8894 df-4 8895 df-5 8896 df-6 8897 df-7 8898 df-8 8899 df-9 8900 df-n0 9092 df-dec 9297 |
This theorem is referenced by: (None) |
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