Download Gödel and Formal Axiomatic Systems Solution Commentary:

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.