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Gödel and Formal Axiomatic Systems Solution Commentary: Solution of Main Problem: 1. One possible proof is: MI Axiom MII Rule 2 MIIII Rule 2 MIIIIU Rule 1 MUIU Rule 3 MUIUUIU Rule 2 MUIIU Rule 4 2. One possible proof is: MI Axiom MII Rule 2 MIIII Rule 2 MIIIIIIII Rule 2 MIIIIIIIIU Rule 1 MIIIIIUU Rule 3 MIIIII Rule 4 3. MU can not be proven in this system. In order to construct a proof (starting with MI), one goal is to produce a string with zero I’s (i.e. a multiple of 3). Axiom MI is a necessary starting place, but it has only one I (i.e. not a multiple of 3). Rules 1 and 4 have no impact on the number of I’s in a symbol string. Rules 2 and 3 impact the number of I’s, but will produce a new symbol string whose number of I’s is a multiple of 3 ONLY IF the number of I’s in the “input” string was a multiple of 3. This cannot occur. Extension 1: In problem #2, MIIIII was proven to be a “theorem” in the MI-generated system. Thus, anything that can be proven starting from MIIIII will be in both the MIIIII-generated system and the MI-generated system. Similarily, we can prove MI to be a “theorem” in the MIIIII-generated system as follows: MIIIII Axiom MIIIIIIIIII Rule 2 MIIIIIIIU Rule 3 MIIIIUU Rule 3 MIIII Rule 4 MIIIIU Rule 1 MIUU Rule 3 MI Rule 4 Thus, anything that can be proven starting from MI will be in both the MI-generated system and the MIIIII-generated system. That is, the two systems are equivalent. Note: Source of this problem is Battista (1982). Extension 2: Neither. If the statement is TRUE, then we have a contradiction of its claim to be FALSE. If the statement is FALSE, the it’s claim is FALSE and the statement must be TRUE. A follow-up question to ask: Is this the same situation as the inability to prove MU as a theorem in the MIU system? Open-Ended Exploration: This problem gets at a meta-examination of possible theorems in the MIU system, rather than staying in the mode of mechanically generating theorems. The resources http://www.norreg.dk/tok/math01.htm and Matos (1997) may prove helpful after students have considered this problem. Teacher Commentary: Stress the similarities and differences between the MIU formal system and Euclidean geometry as an axiomatic system: • The three “undefined” symbols M, I, and U, are analogous to the undefined terms of “point,” “line,” and “plane.” • The single axiom MI is analogous to the Euclid’s Postulates I, II, III, IV, and V. • The production rules are analogous to the standard rules of logical inference used to guide Euclidean proofs. When producing “proofs” of “theorems,” encourage students to explore multiple approaches, while also discussing the associated issue of length of proof. Matos (1997) investigates and constructs numerical bounds for the longest and shortest lengths of a proof for a particular “theorem.” When exploring Problem 3, it is important to explore the implications of what it means to say “a proof of MU” is not possible in this formal system. If one views a “theorem” (e.g. MUIIU or MIIIII) as being “true” in the system, the lack of a proof does not imply that MU is “false.” Rather, it is not possible to establish its truth-value. This provides an opening for a discussion of the ideas of completeness, consistency, and the astounding work of Gödel. To make his actual theorem (1931) understandable, include the two additional “rewordings”: • Gödel’s Theorem: To every ω -consistent recursive class κ of formulae there correspond recursive class-signs r, such that neither ν Gen r nor Neg(ν Gen r) belongs to Flg( κ ), where ν is the free variable of r. • In other words: All consistent axiomatic formulations of number theory include undecideable propositions. • Or: If you have consistency, then you do not have completeness. For more information, see Hofstader (1979), Dawson (1997), Casti & DePauli (2000), Goldstein (2005), http://www.math.hawaii.edu/~dale/godel/godel.html, or http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems. As a good source of a writing project, students can explore any of the following ideas relative to Gödel and formal axiomatic systems: • Proving theorems in this formal system is a mechanical application of four production rules to “true” strings. One can systematically apply the rules, first finding strings that can be generated in one-step from axiom MI, then strings that can be generated in two-steps, etc. The result is a tree such as this: MI MIU MII MIIII MIUIU MIUIUIUIU MIIU MIIUIIU MIIIU MIIIIIIII MUI MIU Etc…the tree continues Create a computer program that will print out one-step theorems, two-step theorems,…, n-step theorems. Make it general enough that you can both change the initial axiom(s) and some of the production rules. (Batista, 1982) • • • • Investigate the meaning and implications of Gödel’s two Incompleteness Theorems, as well as his work regarding first-order logic, the relative consistency of the Axiom of Choice, the Continuum Hypothesis, the limits of formalization, and the concept of a set. Some good resources are Dawson (1997), Casti & DePauli (2000), Goldstein (2005), and Wang (1988). Floyd & Kanamori (2006) claim: Gödel virtually completed the mathematization of logic by submerging ‘metamathematical’ methods into mathematics.” Investigate the meaning and implications of this claim. What events had a bigger impact on mathematics and how it was regarded by others: the Pythagoreans’ discovery of incommensurable magnitudes, the adoption of Hindu-Arabic numerals and numeration system, Zeno’s paradoxes and the use of infinity, Cantor’s transfinite numbers, or Gödel’s theorems? Explain the type of impact and how it could be measured. What exactly does a proof do? Does it justify the “truthfulness” of a statement? For example, Roman Murawski, a Polish mathematician, has stated: Concepts of proof and truth are (even in mathematics) ambiguous. It is commonly accepted that proof is the ultimate warrant for a mathematical proposition, that proof is a source of truth in mathematics. One can say that a proposition is • true if it holds in a considered structure or if we can prove it. But what is a proof? And what is truth? Do you agree or disagree? Explain. (Statement made on http://www.calculemus.org/forum/2/mur01.txt.) Is Euclidean geometry a formal axiomatic system? Are there Euclidean propositions whose truth value cannot be established? How does such a system influence approaches to proof? See Heath (1956), Netz (1999), Mueller (1981), and Kline (1980). Additional References: Batista, M. (1982). “Formal axiomatic systems and computer-generated theorems.” The Mathematics Teacher. March, pp. 215-220. Casti, J. and DePauli, W. (2000). Gödel: A Life of Logic. Persus Publishing. Dawson, J. (1997). Logical Dilemmas: The Life and Work of Kurt Gödel. A K Peters. Floyd, J. and Kanamori, A. (2006). “How Gödel transformed set theory.” Notices of the AMS. Vol. 53#4, pp. 419-427. Goldstein, R. (2005). Incompleteness: The Proof and Paradox of Kurt Gödel. W.W. Norton. Heath, T. (1956). The Thirteen Books of Euclid’s Elements. Vol 1. Dover Publications. Hofstader, D. (1979). Gödel, Escher, Bach: an Eternal Golden Braid. Basic Books. Kline, M. (1980). Mathematics: A Loss of Certainty. Oxford University Press. Matos, A. (1997). “On the number of lines of theorems in the formal system MIU.” Technical Report DCC-97-9. (Available at http://www.dcc.fc.up.pt/~acm/miurep2.pdf.) Mueller, I. (1981). Philosophy of Mathematics and Deductive Structure in Euclid’s Elements. Mineola, NY: Dover Publications. Netz, R. (1999). The Shaping of Deduction in Greek Mathematics. Cambridge University Press. Wang, H. (1988). Reflections on Kurt Gödel. MIT Press.