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Randrew: You said "The problem with the Axiom of Choice is that, using it, one can prove the infamous Banach-Tarski Paradox which states (actually, implies) that one can take a sphere, cut it into six parts, and rearrange the parts to form a new sphere with double the volume of the original (without creating any new empty space inside the object.)". [i remember the statement of the paradox differently, but I never studied it.] This is a reflection of the fact that there are non-measurable sets, that is, sets which do not have a precise volume. This is not a fault of the Axiom of Choice which is merely involved as a convenient way of establishing the existence of non-measurable sets with the necessary properties. The Banach-Tarski paradox is only a problem as long as one persists in trying to treat sets which have no precise volume as if they did have a precise volume. Russell's paradox was an actual contradiction in Frege's development of Cantor's set theory. Thus that naive set theory was destroyed. The Zermelo-Fraenkel Set Theory was constructed specifically to avoid Russell's paradox. As far as we know now, it is consistent. ashleyisachild: Mathematicians do NOT consider the Axiom of Choice to be contradictory! You asked "What is the standard ordering [of the reals]?". As you know, the standard ordering of the integers goes like this: ... < -3 < -2 < -1 < 0 < +1 < +2 < +3 < ... If you divide them by some positive integer such as twenty, you get: ... < -3/20 < -1/10 < -1/20 < 0 < +1/20 < +1/10 < +3/20 < ... If you merge all such orderings together, you get the standard ordering of the rational numbers. In that ordering, (P/Q) < (R/S) means (P*S) < (R*Q) in the standard ordering of the integers, provided that P,Q,R,S are integers and Q,S are positive. Imagine cutting the rational numbers in two with knife in such a way that there are rationals both on the left and on the right and there is no largest rational on the left (smaller) side of the cut. These cuts are the real numbers. If there is a smallest rational on the right side, it is considered to be equal to the real number. The standard ordering of the real numbers is ordering them by how far to the right the cut is in the rationals. So the square root of two is less than pi, because the square root of two is less than (say) +5/2 which is a rational number and +5/2 is less than pi. Mathematicians are well aware that the standard ordering of the reals is not a well ordering. The Axiom of Choice implies that there are well orderings of the reals, but it does not specify any particular one. You asked "How does it do that [assert existence without identity]? But doesn't all of math do that?". Mathematics is based on concepts in which all characteristics of the units (other than the fact that they are units) have been omitted. Eric: I remember from when I studied logic and foundations that there was something called the "no counter-example interpretation" of sentences in analysis. The idea was that one could construct a series of generalized recursive functionals associated with each step of a proof. If someone presented a supposed counter-example to the resulting theorem, then the functionals associated with it would generate a case where the counter-example did not work. So in number theory at least, it should be possible in principle to convert any proof of mere existence into an actual derivation of a number which has the required property. In practice, this calculation would often be impossibly difficult.

Cole: The gravitational field is everywhere, even in space devoid of matter. The gravitational field is the same as the metric (distance or duration) property of space. This is explained in detail in "Gravitation" by Misner, Thorne, and Wheeler, published by W.H.Freeman and Company of San Francisco. The distance from the Earth to the Moon can be measured by triangulation -- using the change in its apparent position as seen simultaneously from two separate points on Earth and trigonometry. Or it could be measured by sending a radio pulse or flash of light to the Moon and measuring how long it takes until you see the reflection. What do you mean by "... I've always understood 'takes up space' to be an essential attribute of matter."? Matter exists in space and it has extension (size). The Pauli exclusion principle prevents two electrons with the same spin from occupying the same space at the same time. Also atomic nuclei repel each other. This limits the extent to which atoms can share the same space. Felipe: You said "... it is entities that have finite qualities and quantities; and 'space' does not qualify as an entity. So, 'space' itself is not 'measurable' without reference to existents.". False. Space also has measurable properties including size. But one must use objects to help delimit the interval to be measured and to either span the interval or send out the radiation which spans the interval. The difficulty is not that space lacks measure without objects, but rather that it has no distinguishable markings without objects in it. pi-r8: I agree with you that The Feynman Lectures on Physics are very good.

Felipe: Yes, there may have been something before the Big Bang, perhaps including other such explosions. In fact, I think that that is the most likely possibility. But one should understand that the only law of physics which forbids creating something from nothing is conservation of energy. Under present circumstances, the new matter would have positive energy as compared with the zero energy of the pre-existing void. This process is forbidden because it increases the total energy, violating the conservation law. But it may be that the Universe as a whole has zero energy, since the positive energy of the matter and various force-fields could be canceled out by the one kind of energy which can be negative which is gravitational potential energy. If that is the case, then the Universe could have appeared from nothing without violating conservation of energy or any other law of physics.

Gabriel_S: You asked "... NAIVE INTERPRETATION ... Please explain.". Let me use the standard example of the stick in the water. A straight stick will appear to bend when it enters the water because the light from the part of the stick below the surface is refracted (bent) as it leaves the water. A naive interpretation of what one is seeing would be that the stick has a bend just at the point where it crosses the surface of the water, because a naif would assume that light always moves in straight lines from the object to his eyes. Actually the naif would not even think in terms of light but would assume that he directly senses the object where it seems to be in his visual field. You said "... you cannot appeal to sensory evidence to establish the validity of other sensory evidence, once you've already entertained the possibility that sensory evidence may be invalid. It's a self-contradiction." I am no longer contending the the senses are invalid. I accepted your argument that one's sensations have a causal dependency on reality and therefore provide information about reality. I have now moved back to the level of interpretation. And I am not contending that one CANNOT interpret one's sensations correctly. What I am saying is that one must use higher reasoning to determine how to interpret them correctly; one cannot simply take correct interpretation for granted. Remember, I said of sense data: "That totality is assumed to be substantially correct for this purpose.".

Gabriel_S said "What assures the unassailability of sensory data? ... The senses are physical instrumentalities that react inexorably according to causal law.". Very good. I grant your point. But as Moose asked "So how do we tell the difference between our mind accurately interpreting sense data and our mind going haywire?". That is, when can we rely on the naive (straight-forward) interpretation of our sensations? Revising what I said before: It is reasonable to presume that the NAIVE INTERPRETATION of one's sensations is correct, but it must be a rebuttable presumption. Gabriel_S asked "Are you going to use sensory data to rebut it?". Yes. The interpretation of ONE piece of sensory data must be evaluated in the context of the TOTALITY of sensory data. That totality is assumed to be substantially correct for this purpose.

Felipe: On the subject of Objectivism's axioms: In the "Can God Exist?, Why or why not?" thread of the "Metaphysics and Epistemology" subforum, TomL said (Apr 11 2005, 09:37 PM) "'Existence exists' doesn't mean what you think it means. It means existence has always existed and always will. It was never created; there has never been a time of non-existence, from which it must have been created. Ever. The universe is eternal, in both directions of time.". That the universe is eternal (towards the past in particular) may very well be true. But I would contend that if "Existence exists." meant what TomL said it meant (which it does not), then it would not an axiom as defined by Objectivism. On page 7 of OPAR, Rand is quoted as saying "[An axiomatic concept is] the identification of a primary fact of reality, which cannot be analyzed, i.e. reduced to other facts or broken into component parts. It is implicit in all facts and in all knowledge. It is the fundamentally given and directly perceived or experienced, which requires no proof or explanation, but on which all proofs and explanations rest.". Correct me if I am wrong, but I take this to mean that a two part test is required for a proposition to be an axiom: 1. It must be known to be true by direct perception. 2. It must be a proposition which is explicitly or implicitly assumed in any proof. These would apply to "Some entities exist, including those entities which I perceive.". This is my understanding of "Existence exists.". It seems clear to me that neither of these properties is possessed by TomL's version. We cannot directly perceive what happened before the Big Bang, so we cannot know that the universe is eternal towards the past. And we certainly do not need to assume that to prove facts about every-day life. This is an example of how some Objectivists read too much into the axioms.

TomL: You said "Nothingness cannot explode into somethingness.". Under present conditions, that is true. But the first moment of the Big Bang was extraordinary. You said "A proof requires that the conclusion be reduced through means of differentiation/integration to the perceptual level.". This is not what "proof" means to a mathematical logician (like me). "not (1 + 1 = 3)" means "from (1 + 1 = 3) one can infer anything", which is true. And I could prove it, if you were willing to wade thru several pages of symbolic logic. My previous impression was that when Objectivists say that "One cannot prove a negative." they were referring to Universal Negatives like "No Man is a Mollusk.". This supposedly on the grounds that one could not examine all men to check whether they were mollusks or not. You said "We know that radishes do not debate, for one. ... These do not constitute proof, because for that you need direct perception.". How do you "know" it, if not on the basis of perception? You said "... anything whatsoever ... Anything less is not 'omni'.". Objectivism correctly emphasizes the importance of context. In particular, the meaning of "all" depends on the context. I am simply choosing a narrower context than the one which you want to use.

I get the impression that most Objectivists disagree with Quantum Mechanics and both Special and General Theories of Relativity. I think that they are over-interpreting their philosophy to draw unjustified conclusions. Basically they are reading the common prejudices which obstruct understanding of these theories into their "Axioms". The expansion of the Universe means that the distances between clusters of galaxies are steadily increasing. So called "empty space" contains various force-fields including the electromagnetic and gravitational fields. Space does not have an end. Although it is not certain, the best available indications are that space is infinite in extent and contains an infinite amount of matter.

Moose asked "... how we know what we sense is real.". Cole said "... the reliability of the senses is an axiom ...". What about: hallucinations, illusions, myopia, ringing-in-the-ears, etc.? It is reasonable to PRESUME that sensations are accurate, but it must be a rebuttable presumption. I think that the criterion has to be whether one's sensation or perception is consistent with the aggregate of facts which one has learned from one's senses and logic.

Cole said of Objectivism: "The essential attribute used to form the definition is that the philosophy was created by Ayn Rand. The fact that the philosophy is also true is a non-essential to the definition.". How can truth be non-essential? Especially to a rational person.

TomL: You said "The universe is eternal, in both directions of time.". Brahma is not supposed to have created the Universe at one time only. Rather He is creating (or re-creating) it continuously, because it could not exist for an instant without Him. It seems plausible to me that the Universe began with the Big Bang about 14,000,000,000 years ago. Why is that not possible? You said "Also, Occam's Razor is false.". Yes. It is merely a heuristic, not a universal truth. That is why I said "suggests". The assumption that "1 + 1 = 3" leads to a contradiction which destroys the mathematical system containing it. If that is not proof that it is absurd, then what is absurdity? What do you consider to be the difference between proof and validation? "omni" means "all", not "infinite to the point of contradiction". Brahma (if He existed) has all power, because He does everything that is done. Brahma has all knowledge (of the present and past of Existence) because they only occurred in His mind.

Life is the product of Evolution. In the course of Evolution, Life becomes more fitted to survival and reproduction in its environment. Airplanes are not separately alive, but neither are your fingernails or even your heart. Our body parts and the tools we create are part of our means of survival. It is our Life, as people who use airplanes, which requires that airplanes be able to fly, not a separate Life of the airplane itself.

The definition of God which I gave in my last message was partly inspired by descriptions of the Hindu god Brahma. Since "Existence exists" is an axiom. It is redundant to suppose that Brahma has created Existence. So Occam's razor suggests that the we dispense with the idea of Brahma. Furthermore, Brahma is a psychological projection of the mental state of a new-born child before he becomes aware that objects continue to exist even when he cannot see them. I was hoping that one of you could find a way to disprove the existence of Brahma. I do not think that the usual arguments against God based on attacking His omnipotence or omniscience will work. Brahma is omnipotent in the sense that He does everything which is done and no force can stop Him from doing what He wants to do with Existence. However, He cannot or would not do anything to limit His own power and He does not create logical contradictions. So his omnipotence is limited in that sense. Also He is omniscient with respect to Existence (in the present and the past). But there may be things which he does not know about Himself and about what He will do with Existence in the future. So the arguments that omniscience is self-contradictory do not apply. If we define a miracle as an exception to the usual rules which Brahma applies to the game of Existence, then I think that He would not make miracles because that would be cheating. Brahma is certainly good because Existence is good. But he is not omnibenevolent. He can and does create things which are evil and which He knows will be evil.

Objectivism correctly holds that Existence is primary, not Consciousness. The concept of God is part of the contrary view that Consciousness is primary. In particular, it is God's Consciousness which is held to be primary. Thus God should be defined as a Mind which makes existence exist by imagining it or by being aware of it. In other words, God creates things by thinking of them. If He ceased to think of them, then they would no longer exist. So reality is like a kind of game of solitaire that God is playing with Himself.

Back in the good old days, we we told not to call ourselves "Objectivists", but rather "students of Objectivism". I am not an Objectivist because I disagree with Ayn Rand on some issues. But I find her writings very illuminating in many cases, so I study Objectivism.

dondigitalia said "A singularity is a phenomena predicted by Einstein's physics in which a large amount of matter is compressed into an infinitely small, infinitely dense entity. Einstein himself had trouble with this part of his theory because he believed nothing in nature could ever be this weird.". Einstein's theory of gravity (with the somewhat confusing name "General Theory of Relativity") assumes that the Special Theory of Relativity holds to a first approximation at every event (place and time) in a free-falling frame of reference. A singularity would be an event where that is not true, in other words, the existence of a singularity would contradict the basic assumptions of General Relativity. So if General Relativity actually predicted a singularity, then it would be contradicting itself. Thus it would be false. Fortunately, it does not predict a singularity. The singularity in the Scharzschild solution arises from the fact that the solution is defined in a contradictory way, IF you try to extend it all the way to the very center. The Schwarzschild solution assumes that there is no stress-energy in the region it covers, but there must be stress-energy somewhere in the central region to generate the non-trivial curvature of space-time outside. General Relativity says that Einstein's tensor (a function of the metric (distance) of space-time and its first and second derivatives) is equal to the stress-energy tensor (mass-energy, momentum, shear, and pressure). It is consistent with any smooth geometry of space-time provided that an appropriate kind of matter exists to produce the equivalent stress-energy. Theorems about the "inevitable" formation of black holes and their central singularities are based on additional assumptions about the kinds of matter that are generating the stress-energy.

AisA asked "Before we go any further, please define what you mean by random.". Remember, my version of the Law of Causality was "The same causes produce the same probability distribution over possible effects ...". So I am not talking about completely arbitrary and unreproducible effects. I am talking about effects which vary within a predictable range and consistently occur with a certain ratio of frequencies. If I generate new instances of the experiment or examine old instances without selecting them by the outcome, then the average over a large number of cases will tend to approach the probability distribution determined by the identity of the causes. Defining randomness generally is a very difficult problem which I think goes beyond the scope of this thread. The point here is that the identity of the causes does not have to determine the precise effects but merely a range of possible effects.

I see four very speculative possibilities for dealing with the central singularity of the Schwarzschild solution: 1. The in-falling particles all combine together into one monster elementary particle. 2. Some unknown state of matter is created at extremely high densities which has a very high negative pressure creating anti-gravity which slows or stops the collapse. 3. Gravity slows time inside the black hole preventing the creation of the singularity until the black hole evaporates due to Hawking radiation. This still leaves the problem of what happens at the last moment when the black hole disappears. 4. The extreme densities force the creation of tachyons (particles moving faster than light-speed) which carry away enough energy to stop the formation of the singularity. - - - - - - - - - - However, the event horizon cannot be removed by such devices. A black hole with a large enough mass (larger than a galaxy) would have only a small tidal effect at its event horizon so that conventional physics would have to apply, not allowing for any escape.

In his second message of Oct 23, 2004, Eric said "The statement 'A well ordering of the Reals exists' is arbitrary mysticism. The well ordering is conjured up by the axiom of choice. It has no referent in reality. It has no use in reality. No one has ever constructed one. No one ever will.". Given a model of Zermelo-Fraenkel set theory, one can form an inner model by limiting oneself to the class, L, of constructible sets. Kurt Goedel showed that L satisfies the axioms of Zermelo-Fraenkel set theory and also "V=L" (an axiom saying that all sets are constructible). "V=L" implies both the Axiom of Choice and the Generalized Continuum Hypothesis. In fact, the entire Universe of the Model, that is the class L, can be well ordered by an explicit formula. Restricting the application of this to the set of real numbers gives an explicit well ordering of the (constructible) real numbers. See "Lecture Notes in Mathematics" #223 "Models of ZF-Set Theory" by Ulrich Felgner, published by Springer-Verlag in 1971. Essentially, one gives a procedure for constructing all constructible sets one at a time. Each ordinal number is interpreted as a code for constructing a set; and each set is the result of such a construction. The real numbers (which are sets) are then ordered by the ordinals that code for them.

dondigitalia asked "Where did Mr. Speicher make this claim?". Excuse me. I should have specified that in my previous message in this thread. Stephen Speicher made the following statements in the "Heisenberg uncertainty principle" thread in the "Physics and Mathematics" subforum: "If by 'randomness' in this context you mean that the exact same non-volitional entity may act differently under the exact same physical circumstances, then that is a violation of the law of causality, and causality is a fundamental axiom of Objectivism." "The Heisenberg uncertainty principle is a separate issue from that of 'randomness on a quantum level.' The assertion of quantum randomness as it is typically made in, say, the standard Copenhagen interpretation of quantum mechanics, is of a metaphysical, not an epistemological nature. It is a destruction of causality itself." "You do not appear to realize that the indeterminism of your 'metaphysical randomness' is a violation of both identity and causality. To exist is to be something specific, and an entity's identity is the sum total of all of its attributes and characteristics. When we say that an entity acts in accord with its nature, we mean that the entity's attributes and characteristics determine, in any given set of physical circumstances, what an entity can do. The action possible to an entity is caused and necessitated by what the entity is." "For an entity to act against its nature would be a contradiction of its identity; it would be saying that, in spite of the specific attributes and characteristics possessed by the entity, it would act otherwise. According to your notion of metaphysical randomness, an entity could act in two contrary ways under the exact same physical circumstances, and that would imply the existence of two contrary attributes being part of the entity's nature, and that would be a contradiction." "Identity is the totality of attributes and characteristics possessed by an entity, and such attributes and characteristics cannot be contradictory. If an entity were able to move left in some particular physical circumstances, and yet able to move right in the same physical circumstances, such contradictory actions would be derived from contradictory aspects of the entity's nature. But contradictions do not exist, A is A, and the only action possible to an entity, in any given physical circumstances, is that single action which is caused and necessitated by its nature. Choice is that single action possible to a volitional consciousness, and it is that attribute which permits freely selecting from among alternatives. And, unlike the notion of 'metaphysical randomness,' causality is operative throughout." I did not post my message in that thread because I wanted to discuss this issue in the context of the Law of Causality rather than in the context of Quantum Mechanics. Consequently, I put my new thread in the "Metaphysics and Epistemology" subforum.

The Law of Causality is a corollary of the Axiom of Identity. Stephen Speicher has claimed in another thread that the Law of Causality is incompatible with the randomness of quantum mechanics. I disagree. I would formulate the Law of Causality as follows: The same causes produce the same {probability distribution over possible} effects [regardless of when the event occurs, or where, or in what orientation, or how fast it is moving (below light-speed), or in which direction it is moving]. The part in braces {} takes account of the nondeterminism of radioactive decay, free will, and quantum uncertainty. The part in brackets [] incorporates the special principle of relativity, a part of the special theory of relativity. I do not think that the Axiom of Identity forbids the part in braces.

IdentityCrisis asked "... what is the mathematical definition of infinity (or what are the definitions, if more than one)?". A set is infinite if it is not finite. There are several definitions of finite. These are equivalent if the 'Axiom of Choice' is true. 1. A set is finite if it can be placed into a one-to-one correspondence with the set of natural numbers (non-negative integers) less than some specific natural number (the cardinality of the set). 2. A set if finite if it has all properties which can be proved by mathematical induction beginning with the empty set and adding one new element at a time. 3. A set is finite if it can be given a total ordering which is both well ordered forwards and backwards. That is, a set is finite if every non-empty subset has both a least and a greatest element in the subset (these are the same if the subset is a singleton). 4. A set is finite if every function from the set one-to-one into itself is onto. 5. A set is finite if every function from the set onto itself is one-to-one.