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The outlook includes constant re-assessments of the theory -- of any theory -- and analysis of the errors involved as one updates and improves it. Tarski's Indefinability Theorem According to this theorem no sufficiently powerful language is strongly-semantically-selfrepresentational.

For instance, any arithmetical truth cannot be defined by arithmetics. Tarski's theorem is not directly about mathematics but about the inherent limitation of any formal language that is strongly-semantically-self-representational when it contains predicates and function symbols defining the semantic concepts specific to that language. Even in computer science one cannot create a complete and consistent finite list of axioms, or an infinite list.

Each time one adds a statement as an axiom; there are other correct statements that cannot be proved, even with the new axiom. Moreover, if the system can prove that it is consistent, it is not. As might have been expected, this idea has been much debated by mathematical philosophers: v How can a theory be both correct and unprovable? Not all axiom systems satisfy these hypotheses, even when these systems have models that include natural numbers as a subset.

Another limitation applies only to systems that are used as their own proof systems. Theorems are computably enumerable: one can write a computer program that will eventually generate any valid proof. One can then ask if it has the stronger property of being recursive: Can one write a computer program to definitively determine if a statement is true or false? His theorems, however, are confined to sufficiently strong axiomatic systems, i.

Nevertheless, the essence of these issues is much more complicated, as indicated below. Undecidable Statements: A statement is neither provable nor refutable. Hence, some scholars resort to the concept "independent".

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However, that concept is also ambiguous. Meta-Language: A sufficiently developed language cannot represent its own semantics. Any metalanguage includes primitive notions, axioms, and rules absent from an object language. Theorems provable in a meta-language are not provable in the object language. If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent.

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Sets of expressions are considered as coded as sets of numbers. For various syntactic properties such as a formula, a sentence, etc.

And any computable set of numbers can be defined by some arithmetic formula. There are various additional theorems and sub-theorems I do not intend to expand on them in these pages, for these pages are mainly written for the general reader, without resorting to any mathematics. We therefore move next to elaborate on some more practical domains in the mined fields of verifiable scientific theories and proofs.

It may protect one from making mistakes, prior to and aft writing down any mathematical equation. This may be best demonstrated by the most dramatic event in the history of science; when Einstein lost trust in the most verified gravity field equations of general relativity3.

Investigations in Philosophy of Cosmology

For good reasons stated above Einstein's tensorial gravity field equations were flawless, despite the fact that a cult has evolved to demand adding this factor to fit its ulterior wishes to operate under some set of philosophical prejudices. Most important, the failure to acknowledge this is not only self-delusion, but a failure to attend to the philosophical base of physics, a failure that leads to empty scientism. Space-Time vs. Symmetry-Asymmetry The essence and fundamental meaning of curved space-time, symmetry-asymmetry, statistics and probabilities are understood only by a portion of scientists.

Differential Equations vs. Observations Time-symmetry, reversibility and the symmetrical laws of physics serve us as key tools to better understand nature, despite the limitations mentioned above. And it is only by combining conceptual, reversible, analytic time-symmetry with factual, aggregated, observational time-asymmetry, or by combining reversible equations with a priori known, factual, initial and boundary conditions, that one may,.

Mathematics, gravitation, symmetry-asymmetry, aggregation, time and meaningful sentences are coupled. While some minor reservations are justifiable, there is an overwhelming "word of evidence", derived from physics, the languages and the studies of linguistics, cybernetics, information and mathematics that fortifies this contention. A complete and consistent set of axioms for all mathematics is impossible.

A Debate Over the Physics of Time

Yet, consistency, by itself, is a two-sided issue. When we find that other sets of phenomena are incompatible with that theory, we tend to either generalize or modify it, or failing that, seek an alternative one.

Yet, Einstein did not trust some semi-hidden aspects of his own general theory of relativity2. See Table of Contents,. Namely, the foundations of thermodynamics should NOT be a separate, stand-alone, theory of science; it should NOT be fragmented from the rest of physics. It constitutes an integral part of gravity-induced, interconnected, unified physico-philosophical knowledge. DO armies of gate guards defile our temple of knowledge? Einstein's general relativity and Hilbert spaces may have been lost to humanity by today's red tape that defiles our temple of theoretical physics?

WHAT kind of science is so smuggled into present and future knowledge? The reason why men enter into society is the preservation of their property.

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Why gravity is the most fundamental integrator of all studies? Volume I demonstrate this fact within the studies of physics, astrophysics, astronomy, cosmology and philosophy, while Lecture 5 extends it to the origin of the sun, earth, life, hominids, writing and civilization. We next extend it to gravity- induced sociology and the philosophy of natural selection: The fight for property-domains-resources, the socio-rules of time-grammar, gravity-induced territorial wars, city and village structures, transportation systems, aspirations, religions, theology and the arts.

We observe this phenomenon in both the animal and human domains of socio- biology. One species, or one sex, or one race, may control a larger quantity of resources, say, scarce water ponds, than the other, until a small percentage of the population monopolizes key resources. Gravity-induced hills and mountains control castle and village structures and especially tall structures may become worship temples of the super- natural.

RECAP: Domain-Property and Envy drive competition, company ads, venture capital, TV and academic rating, political candidates, religious rivalry and key human traits, sometimes stronger than death. As history testifies, gravity-induced territoriality is often developed into nationality, imperialism and territorial conquest wars. Since biogeochemical evolution causes key natural resources to be distributed non-uniformly in the spatial and temporal coordinates of the biosphere, resource monopolization develops in all levels of socio-biological systems.

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RECAP from previous Lectures: The first local "aggregates" of matter in the solar system have already contained gas, supernovae debris, dust, rocks. The atmosphere and geological strata are ordered-selected by specific gravity of their chemical components. It was gravity that put the early components of living systems at one and the same strata for life to begin.

Gravity periodicities around the sun emerge as prime sources of order via time and the arrow of time [Lecture 1]. Animals are also 'conscious' of the commonly-shared "up- down" surroundings, and about "weight" and acceleration. Plants also harbor innate sensors that guide them in which direction to grow even on a mountain slope. Additional examples range from gyroscope inertial changes, balancing machines, heading indicators and gravity-induced changes involving the fluid in the vertebrate inner ear to a crystallizing suspension of organic spheres in water.

Gravity-induced Plant Growth and Animal Behavior The sensing devices which plants and animals use for "gravity perception" "gravity receptors", "g perception", "bio- accelerometers", "gravity- induced biological clocks", etc. But what we already know justifies the central role we claim gravity plays in living systems and society.

As most high-growing trees they align their growth under the control of the gravity vector. If a growing higher plant is displaced with respect to the "upright" position, some tens of minutes later it will adapt its growth in such a way as to restore its original orientation in coincidence with the gravity vector.

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If it is displaced only briefly and then restored to its original orientation well before the growth response can set in, it still responds to that displacement. Gravity-induced orientation-adaptation of an organism may occur when an organism orients itself by a gravity-induced gradient of density differences or hydrostatic pressure. Small organisms including all bacteria may have no means for sensing gravity, but they are affected by it. Animals low on the evolutionary scale characteristically exhibit innate patterns that depend little on learning, and have a lesser adaptability to changes in the environment.

While the latter show some signs of innate The evolutionary origin of the so-called "innate ideas" may, therefore, be external rather than internal. Even the computer page is ordered by up, bottom, left, right margins and footnotes.

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And it takes time to read them and to comprehend them; the time that always advances from past to future; the time that we term Linguistic Arrow of Time. Similarly, gravity-induced ordering of shelves, files, documents and books in a library introduces order into our life. Gravity- Chained Law, Seconds, Minutes, Days, Years Whole civilizations religions constructed on gravity-induced orientability, concepts, spirituality, art, music, home structure, wells, canals, trade roads, agriculture, and even law enforcement by hanging.

Such gravity-induced cultural phenomena may proceed beyond grammar and social order into the spiritual and hierarchical cultures, and have been domains: High priest, low deck, high commissioner, and highness, heaven, lofty, go to the bottom of the subject, etc. Biological Clocks The origin of temporal behavior in animals can be traced back in time and out to external physical influences.

Gravity and geophysical periodicities thus emerge as prime sources of order and information in all non-living and living systems. Animals low on the evolutionary scale depend little on learning, and have a lesser adaptability to changes in the environment. Consequently, their dependence on heredity-geophysical-gravitational origins is high in proportion to animals high on the evolutionary scale.

The vestibular system sends signals primarily to the neural structures that control our eye movements, and to the Cerebellum-muscles system that keeps us upright and helps maintain clear vision-orientation vs. Reproduction is also impaired and the immune cells cannot differentiate into mature cells in low or zero gravity conditions attainable in space craft. The size of a biological cell depends on the local gravity vector — its size increases in larger local Bone cells must attach themselves to something and will die if they cannot.

Without the gravity force- field they float around and perish. Earlier life forms in the oceans were smaller and had a jellyfish-like configuration. Thus, without strong skeletons, land animals cannot evolve.