For me, there is only one logic that we rational human beings are able to accept and appreciate, and that is the mathematical logic of ZF theory. But in the last century we found that ZF theory is not in a position to provide all that we want, and went in search of a new mode of thinking and got one which we called meta mathematics. My question is: if we can put the unambiguous logic of ZF theory on paper, why can't we do the same with meta mathematics. This paper is my feeble attempt in that direction.

1. Sentient Arithmetic and G¨odel’s Theorems
K. K. Nambiar
School of Computer and Systems Sciences
Jawaharlal Nehru University, New Delhi 110067, India
(Received June 1995)
Abstract—Sentient Arithmetic is defined as an extention of Elementary Arithmetic with three
more derivation rules and the Incompleteness Theorems are derived within it without using any
metalanguage. It is shown that Consistency cannot be chosen as an axiom. Addendum 2015
at the end gives details of my current views.
Keywords—Sentient Arithmetic; Consistency; G¨odel’s Theorems.
1. INTRODUCTION
Since the discussion here is about some of the subtle aspects of mathematical logic [1-3], a brief
but complete definition of Sentient Arithmetic (SA) is given first. We follow the conventional form
of definition of an axiomatic theory, except that axioms are listed as part of the derivation rules.
The purpose of this paper is to show that it is possible to prove the incompleteness theorems of
G¨odel entirely within SA, without using any metalanguage.
1. Symbol Schema
1. Logical symbols
∨ ∧ ∼ , ) ( ∀ ∃
Here, ∨ stands for Or, ∧ for And, ∼ for Complementation, ∀ for For All, and ∃ for There Exists.
2. Variable symbols
x, x1, x2, · · · , y, y1, y2, · · · , z, z1, z2, · · ·
These symbols are similar to the variables of mathematics.
3. Label symbols
a, a1, a2, . . . , b, b1, b2, . . . , c, c1, c2, · · ·
These symbols are similar to the arbitrary constants of mathematics. Eventhough they are not
available in conventional logic, we have a good reason for introducing them. By defining them
here, we have the flexibility to dispense with the quantifier symbols when convenient: ∀xP(x)
can be written as P(x) and ∃xP(x) as P(a).
4. Constant symbols
0, 0 , 0 , 0 , · · ·
usually written as
0, 1, 2, 3, · · ·
5. Equality symbol
=
defined by the derivation rules given later.
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2. 6. Arithmetical symbols
0 + ×
The symbol is read as successor.
7. Defined symbols
New symbols can be defined in terms of the above symbols as done later in the text. Such
definitions can be useful for creating new logical symbols and also for abbreviating long formulas
and large numbers.
2. Term Schema
1. 0, a, x, a1, x1, a2, x2, · · · , b, y, b1, y1, b2, y2, · · · , c, z, c1, z1, c2, z2, · · · are terms.
2. (u) + (v), (u) × (v), (u) are terms if u and v are terms.
Note that the symbols u and v are used only to describe the schema.
3. Formula Schema
1. u = v is a formula, if u and v are terms.
2. If p and q are formulas, then (p) ∨ (q), (p) ∧ (q), and ∼ (p) are formulas. Note that the
symbols p and q are used only to describe the schema.
3. ∀s p(s) and ∃s p(s) are formulas, if p is a formula and s is a variable in p(s). The logical
symbols ∀ and ∃ (called quantifiers) never occur by themselves. They are always accompanied
by a variable and appears as ∀s or ∃s. Note that s is used only to describe the schema.
4. Definition Schema
1. F(x): We assume that the formulas of SA can be enumerated. The function F(x) gives
the xth
formula in the list. In the formation of the formulas we assume that more than one
complementation at a time is not allowed, since it does not serve any purpose and will merely
complicate our discussion. We will refer to F(x) as the formula stored at address x. In the list,
address 0 is reserved for a special formula G given later.
2. ¯x: The address at which ¯F(x) is stored we call ¯x. Thus F(¯x) = ¯F(x). It is easy to see
that ¯x is a primitive recursive function of x, and ¯¯x = x. Roughly, a function is recursive if it can
be programmed.
3. P(x, y): The primitive recursive predicate (a very long formula) which says that the
formula F(y) is a proof of the formula F(x).
4. D(x): An abbreviation for ∃y P(x, y) which says that F(x) can be derived. It is not a
recursive predicate.
5. F(x) ⇒ F(y): The same as ¯D(x) + F(y). When the context makes it clear, we will use
+ instead of ∨, as is common. Similarly we omit ∧ whenever the omission is obvious.
6. F(0): The formula ∼ ∃yP(0, y) is stored at address 0. F(0) says that F(0) cannot be
derived. We will use the symbol G for ∼ ∃yP(0, y) and for uniformity F(g) for F(0). Note that
G can also be written as ¯D(g) and ¯G as D(g). Observe that keeping the formula ∼ ∃yP(0, y) at
address 0 in no way affects the recursive nature of F(x).
7. F(c): The formula ∼ ∃x D(x)D(¯x) has to appear some where in our list, we call that
address, c. We will use the symbol C for ∼ ∃x D(x)D(¯x). C says that it is impossible to derive
both F(x) and ¯F(x). C is read as consistency and, ¯C as contradiction.
5. Derivation Schema
In the following a single line in the statement of the rule means that it can be freely introduced
anywhere in a derivation, in short, they are axioms. The used here is a rotated turnstile symbol
with the meaning that the following line can be derived from what preceeds. The meaning of ⊥
should be obvious. Note that these symbols are used only to describe the schema.
1. Sentential rules
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3. Commutation rule
a) p + q b) pq
q + p qp
Distribution rule
a) p(q + r) b) p + qr
⊥ ⊥
pq + pr (p + q)(p + r)
Identity rule
p + ¯C
⊥
p
Complementation rule
a) p b) p¯p
⊥
p ¯C
These derivation rules have been obtained from the definition of boolean algebra. The omission
of the axiom p + ¯p is intentional here, since SA has no use for the law of the excluded middle.
Detachment rule
p
p ⇒ q
q
2. Predicate rule
a) ∀x p(x) b) p(u)
∃x p(x) ∃x p(x)
3. Equality rule
a) u = u b) u = v
p(u, u) ⇒ p(u, v)
4. Peano rule
a) ∼ ∃x (x = 0) b) u = v c) p(0)
∀x [p(x) ⇒ p(x )]
u = v
∀x p(x)
5. Addition rule
u + 0 = u u + v = (u + v)
6. Multiplication rule
u · 0 = 0 u · v = u · v + u
7. Sentient rules
Validity rule: This rule essentially gives a syntactic definition of truth.
D(u)
F(u)
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4. Introspection rule: This rule says that if you have a legitimate derivation of F(u) visibly in
front of you, you can conclude that D(u) is true.
...
F(u)
D(u)
Contradiction rule: This rule says that any formula that leads to a contradiction cannot be
derived.
F(u) ◦ assumption
...
¯C
C ⇒ ¯D(u)
It is legitimate to use both the validity rule and the introspection rule under the assumption
of the contradiction rule. This rule we may call no-proof by contradiction. The usual proof by
contradiction is not allowed in our theory.
The schemas given above define Sentient Arithmetic. If we omit derivation rules 7 we get
the Elementary Arithmetic of G¨odel. There are no seperate axioms in SA, they are embedded in
the derivation rules. A theorem in SA is the last line in a derivation. A derivation in which the
last line is ¯C is called a virus.
2. G¨ODEL’S THEOREMS
Using the definitions given above, we can prove the incompleteness of SA entirely within
SA.
FIRST INCOMPLETENESS THEOREM. C ⇒ ¯D(g) ¯D(¯g)
Proof of C ⇒ ¯D(g)
1. G ◦ assumption
2. D(g) ◦ introspection rule on line 1
3. ¯D(g) ◦ definition of G at line 1
4. ¯C ◦ from lines 2 and 3
C ⇒ ¯D(g) ◦ contradiction rule
Proof of C ⇒ ¯D(¯g)
1. ¯G ◦ assumption
2. D(g) ◦ definition of ¯G at line 1
3. G ◦ applying validity rule on line 2
4. ¯C ◦ from lines 1 and 3
C ⇒ ¯D(¯g) ◦ contradiction rule
First Incompleteness Theorem immediately follows.
SECOND INCOMPLETENESS THEOREM. C ⇒ ¯D(c) ¯D(¯c)
Proof of C ⇒ ¯D(c)
1. C ◦ assumption
2. C ⇒ ¯D(g) ◦ first incompleteness theorem
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5. 3. ¯D(g) ◦ detachment rule on lines 1 and 2
4. G ◦ ¯D(g) at line 3 is the definition of G
5. D(g) ◦ applying introspection rule on lines 1 and 4
6. ¯C ◦ from lines 3 and 5
C ⇒ ¯D(c) ◦ contradiction rule
Proof of C ⇒ ¯D(¯c)
1. ¯C ◦ assumption
C ⇒ ¯D(¯c) ◦ contradiction rule
Second Incompleteness Theorem immediately follows.
3. CONCLUSION
From our derivations it should be clear that SA abhors a contradiction. Then the ques-
tion arises, why we should not introduce consistency itself as an axiom of SA. If we do that,
unfortunately, viruses invade SA and it collapses, as shown by the argument below.
1. C ◦ new axiom introduced
2. D(c) ◦ applying introspection rule on line 1
3. C ⇒ ¯D(c) ◦ second incompleteness theorem
4. ¯D(c) ◦ detachment rule on lines 1 and 3
¯C ◦ from lines 2 and 4
The conclusion is that C cannot be introduced as an axiom of SA. We may define F(x) as a
metastatement of SA, if both F(x) and ¯F(x) leads to a contradiction. Obviously, a metastatement
cannot be chosen as an axiom of SA.
If what we have discussed here is sensible, we can divide the statements in SA into four
classes. F is a contradictory statement if a derivation exists for both F and ¯F. F is an arithmetical
statement if a derivation exists for either F or ¯F, but not both. F is a mystic statement if a
derivation exists for neither F nor ¯F and it is not a metastatement. As stated above, F is a
metastatement if a derivation exists for ¯F when F is assumed and a derivation for F exists when
¯F is assumed.
To judge the quality of an axiomatic theory we make the following definitions. A theory is
sound if there are no contradictory statements in it. A sound theory is rational if there are no
mystic statements in it. We can now state the belief and hope that can be entertained about
Sentient Arithmetic, in terms of these concepts.
Arithmetical Faith: SA is sound
Arithmetical Hope: SA is rational
Our discussion here clearly shows that it will be fatal for SA to convert the faith into an axiom
and unrealistic to take the hope as a fact.
SUMMARY AND COMMENT ADDENDUM 2015
Reproduced below are my comments from the Web, essentially summarizing what I have
said above. Slight change in terminology is introduced for clarity.
I first enhance Zermelo-Fraenkel (ZF) theory a little and call it Sentient ZF theory (SZF
Theory). The definition of SZF will show that it is more knowledgeable about its formulas than
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6. ZF.
Definition of Sentient ZF Theory:
SZF theory is an enriched form of ZF theory except that the Law of Excluded Middle is
prohibited. Also, three additional derivation rules, as defined below, are allowed. These rules,
fabricated by me, are the ones used by G¨odel in his metamathematical reasoning. In the statement
of the derivation rules, means that the following line can be derived from what precedes, and
⇒ stands for ‘implies’.
Notations:
P(X,Y): The primitive recursive predicate (a very long formula) which says that the formula
Y is a proof of the formula X. (Note: Roughly, recursive = programmable.)
D(X): An abbreviation for, ”there exists a Y such that P(X,Y)”, which says that X can
be derived. It is not a recursive predicate. We use the symbol C for ‘Consistency’ and ¯C for
‘Contradiction’.
Augmented derivation rules of SZF theory:
1. Validity rule: This rule says that, from D(F) you can infer F.
D(F)
F
2. Introspection rule: If you have a valid derivation of F, you can infer D(F).
:
:
F
D(F)
3. Contradiction rule: If F leads to a contradiction, then you can infer C⇒ ¯D(F).
F
:
:
¯C
C ⇒ ¯D(F)
We give these derivation rules only for our present discussion. The exact logical notations
for them and their use can be found in the paper above.
The three inference rules allow SZF theory to critically observe the working of ZF theory
and make logical conclusions from it (hence the name sentient). In what follows most of our
discussion and conclusions are about the ZF theory. It is assumed that the vocabulary and
grammar of ZF and SZF are the same. The truth in ZF theory we will call ‘relative truth’ and
the truth in SZF the ‘absolute truth’.
Here is the generally accepted visualization of axiomatic ZF set theory: The grammar of
ZF theory is supposed to give us all the sentences we will ever need for a rational discussion of
set theory, including the concept of infinity in it. The fond hope of the set theorist was that
every formula in it can be derived from the initially chosen set of axioms as either a theorem or
a falsehood. If it turns out that a formula is neither a theorem nor a falsehood, the set theorist
marks it as a candidate axiom, which for the purposes of our discussion we will call an ‘enigma’.
Thus, the set theorist thought that, if you ignore axioms, there would be three kinds of formulas
in ZF theory – theorems, falsehoods, and enigmas. All this changed when G¨odel came out with
his seminal paper in 1931, when the enigmas got converted into undecidable formulas of ZF
theory. But, that will be getting ahead of the story.
G¨odel has shown that it is possible to construct a formula G in ZF theory for which both
the following statements are absolutely true.
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7. 1. Consistency implies that a derivation of G does not exist.
2. Consistency implies that a derivation of ¯G does not exist.
From this it follows that neither G nor ¯G can be introduced as an axiom of ZF theory, because
of the consequent contradictions. Thus, it is clear that the undecidable G cannot be classified
as a possible axiom of ZF. What shall we call the G¨odelian sentence G? Some have called it a
paradox, and yet others have called it an absurdity. I am going to call it an ‘introversion’. What
shall we call the rest of the undecidable formulas of ZF which are not introversions? I will call
them ‘profundities’. From all this, it will not be unreasonable to say that, apart from axioms,
there are four types of sentences in ZF theory, namely, theorems, falsehoods, introversions, and
profundities.
There is a world of difference between introversions and profundities. Continuum hypothesis
is a significant profundity of ZF and it needed considerable research by G¨odel and Cohen to
discover that fact. The crucial difference between an introversion and a profundity is that while
a profundity can be chosen as an axiom of ZF theory, we cannot do the same with an introversion.
If we choose an introversion as an axiom, ZF theory will be riddled with contradictions resulting
in its collapse. From all this, my contention is that G¨odel’s 1931 result proves nothing more than
the theorem below.
G¨ODEL’S FATALITY THEOREM: There are absurd undecidable formulas in ZF theory
called introversions, any of which if chosen as an axiom can be fatal to the theory.
From the work done by G¨odel and Cohen in investigating the continuum hypothesis, we can
state the following theorem regarding the incompleteness of ZF theory.
G¨ODEL-COHEN INCOMPLETENESS THEOREM: There are significant undecidable for-
mulas in ZF theory called profundities, any of which can be chosen as an axiom without creating
contradictions.
What I have said here should in no way be construed as belittling the importance of the
fatality theorem. It is important to note that an introversion as an axiom is fatal to ZF theory,
and consequenly to SZF theory also. An important theorem proved by G¨odel shows that the
statement ‘ZF theory is Consistent’ is an introversion. If not anything else, this should prevent
a reckless set theorist from starting off his theory with the first axiom stating that his theory is
consistent.
According to my formulation, a formula is undecidable, if it is either an introversion or a
profundity. I do not believe that any one holding the views as I have given above will ever
maintain a position that Riemann Hypothesis (RH) is an undecidable problem. It is difficult
for me to feel that the elegant simply stated RH is a profundity, much less an introversion. I
am making these statements only because there are some mathematicians who claim that RH
could be undecidable. My earnest suggestion to the mathematician is that he should ignore the
undecidability concept of the logician in his daily research work. What happened in the case
of problems like four-color conjecture and Fermat’s last theorem supports my position in this
matter.
Parenthetically, I may add that G¨odel did not talk about ZF theory, but had used a tiny
fragment of Principia Mathematica (PM) to prove his theorems.
I recognize that my views here are considerably different from that of conventional mathe-
matical logic, but I am convinced that this is a very sensible point of view.
REFERENCES
1. R. R. Stoll, Set Theory and Logic, W. H. Freeman and Company, San Francisco, CA, (1963).
2. E. Mendelson, Introduction to Mathematical Logic, D. Van Nostrand Company, New York, NY, (1979).
3. J. R. Shoenfield, Mathematical Logic, Addison-Wesley, Reading, MA, (1967).
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