Infinity may be defined as a quality or state of being limitless, endless, unbounded, immeasurable, or innumerable. The infinite is something for which a complete description of all its qualities, elements, or parts is impossible, because all of these qualities, elements, or parts can never be completely enumerated. The infinite is something which can never be completely itemized or catalogued as a totality. Any attempt to catalogue or enumerate all the elements of the infinite can never be completed. The infinite surpasses or transcends any attempt at complete description.
While the enumeration of all the elements of the infinite must be incomplete, the infinite in all of its elements may be regarded as complete. The infinite in all of its elements may be regarded as a totality. The infinite may be regarded as a whole realm of elements, aspects, or qualities which form an endless or absolute unity.
Thus, the infinite may be viewed either in terms of incompleteness or completeness. The infinite may be viewed either as an object of knowledge (in which case it must be incomplete), or as something which transcends knowledge (in which case it may be complete). The infinite may be viewed as something which either frustrates or inspires the attempts of the finite mind to comprehend it.
Infinity cannot be adequately represented by any finite concept of quantity. Any attempt to represent infinity by means of a finite concept of quantity is insufficient to produce an adequate representation. A finite concept of quantity may have infinity as its object, or may have any other quantity as its object. An infinite concept of quanity may also have infinity as its object, or may have any other quantity as its object. But infinity cannot be adequately represented by finite concepts. Infinity can only be adequately represented by concepts which are themselves infinite. Concepts which are finite, and which have infinity as their object, are only limited representations of something which cannot be adequately comprehended by finite methodology.
A concept of a finite object is not an adequate concept of an infinite object. A concept of an infinite object which attempts to represent that object as a single finite object cannot be an adequate concept of that infinite object.
It is impossible for the finite mind to produce an infinite number of concepts. Moreover, it is impossilbe for the finite mind to produce any concept that is infinite. A concept may be defined as infinite if it has an infinite number of objects. A concept may be defined as finite if it has a finite number of objects. The finite mind can only produce finite concepts, and thus it may be impossible for the finite mind to produce an adequate concept of something which is infinite. Infinite concepts can only be produced by an infinite mind. Thus, only an infinite mind can produce an adequate representation of infinity.
An adequate concept may be defined as one which is equal to the dimensions of its object, or one which is fully sufficient to comprehend its object. An adequate concept of infinity must therefore refer to something which has an infinite number of elements, qualities, or parts. Each of the elements or qualities of infinity must be an object of that adequate concept.
Infinite numbers of concepts may be produced by an infinite number of finite minds, but no single finite mind may produce infinite numbers of concepts. Infinite numbers of finite concepts may produce an infinite number of representations of infinity, but a single, unified concept of infinity must have an infinite number of objects if it is to be an adequate concept. Only an infinite mind can produce such a concept.
For the infinite mind, there are an infinite number of possible concepts and an infinite number of possible objects of thought. The infinite mind can combine an infinite number of concepts, in an infinite number of ways. The finite mind can combine only a finite number of concepts, in a finite number of ways.
Any concepts of infinity which are produced by a finite mind can only be enlarged upon to produce more adequate concepts. This process by which concepts are enlarged upon, or more complex concepts are developed, may continue indefinitely, but can never produce a fully adequate concept of infinity. An adequate concept of infinity surpasses or transcends the limits of the finite mind. Thus, for the finite mind, the infinite is a transcendent category or mode of being. For the finite mind, the infinite transcends conceptual knowledge. If the finite mind is to attain any adequate knowledge of infinity, this knowledge cannot be attained by using concepts, but must be attained by using other instruments of knowledge.
If a fully adequate concept of infinity is impossible for the finite mind, does infinity exist? Can infinity be proven to exist by any form of conceptual argument? If we are to attempt to prove that infinity exists, we must know what we mean by the term 'infinity.' But infinity cannot be adequately represented by any concept, and thus cannot be proven to exist by any form of conceptual argument. Any logical proof of the existence of infinity can only argue for the existence of one or another inadequate concept of infinity. Thus, any supposed proof of the existence of infinity which uses conceptual argument can only prove the existence of something other than infinity.
Any concept of infinity as a numerical quantity is paradoxical, because there is no quantity which can be assigned a real number corresponding to infinity. If a numerical quantity could correspond to infinity, then there would be a real number which could correspond to this quantity. But any real number which could be chosen to represent this quantity could only be enlarged upon by another real number. Thus, infinity is not a qunatity which can be assigned a real number.
An infinite number is an indefinite number. An indefinite number has a range of possible values, and an infinite number has an infinite range of possible values. Infinity cannot be assigned a single real number, but is only something which may be approached by a sequence of numbers. An infinite number may be an infinitely large or an infinitely small number. To say that infinity is something which may be approached by a sequence of numbers is to say that the sequence of numbers is limitless.
If infinity exists, then that which is infinite may be real. If infinity does not exist, then that which is infinite may be unreal. If the finite mind can comprehend every aspect of reality, then the infinite cannot be real. If reality is infinite, then the finite mind cannot comprehend every aspect of reality. Only an infinite mind can comprehend every aspect of infinite reality.
If infinite reality includes finite as well as infinite aspects, then the finite mind may comprehend some aspects of infinite reality. On the other hand, if every aspect of infinite reality is infinite, then the finite mind cannot comprehend any aspect of infinite reality.
If reality has an infinite number of finite or infinite aspects, then the finite mind cannot comprehend every aspect of reality. However, even if there are a finite number of aspects of reality, and every aspect of reality is finite, the finite mind may still not be able to comprehend every aspect of reality.
Aristotle distinguishes between actual infinity and potential infinity. A series of numbers cannot actually stop at infinity, because an infinite series of numbers never stops. Infinity may not actually exist at a particular point in time, but may potentially exist over a period of time.1
If the universe is infinite in size, then it is impossible to determine any specific location for the center (or periphery) of the universe. If the universe is infinite in size, then any point in space may be designated as the center (or periphery) of the universe.
If the universe is infinite in size, it cannot be greater or smaller in size than it already is. If the universe is infinitely large in size, then it can never be completely traversed. If the universe is infinitely large in size, then an infinite amount of time would be required to completely traverse it. If the universe is infinitely large in size, then there is no point in time at which a journey across the whole universe could be completed. If the universe is infinitely large in size, then time and space are infinite, and regardless of how far an observer has travelled across the universe, there is still more time and space to travel farther.
If time is circular, then time may not be infinite. If time may continue long enough for an observer to return to the same point in time, then time may be circular and not infinite.
If time is not circular and not infinite, then there must be a beginning and end of time, and a beginning and end of the universe. If time as duration had a beginning but has no end, then by definition it must be unlimited or infinite in duration. If time as duration has an ending, then it must have had a beginning. If time as duration has neither a beginning nor an ending, then it must either be infinite (having always existed, and having no ending) or non-existent (having never existed, and having no beginning or ending).
What is beyond infinity? If infinity is without limits, then nothing can surpass infinity. The concept of infinity as something which is insuperable or which cannot be comprehended in its totality may thus be a limiting concept. All finite things or beings may be limited by their inability to attain or comprehend infinity.
Infinity may have many forms or dimensions. Infinity may have mathematical, logical, physical, spatial, and temporal forms or dimensions. Infinity may also have personal, social, psychological, aesthetic, moral, or religious dimensions.
If the infinite includes, or is composed of, an infinite number of finite things or beings, then finite things or beings may reveal (or express) the principle of infinity. The infinite may be revealed by the finite, and the finite may have within itself the principle of the infinite. The infinite may combine finite things, or finite elements, or finite beings, in an infinite number of ways.
Infinity may be a state of monotony. If something is infinite, it goes on and on, ad infinitum. Infinity is something that does not stop. Thus, infinity may be regarded as a form of continuity, rather than as discontinuity. Infinity has a continuity which is endless.
Concepts which are produced by the finite mind may be expressed (or may have meaning) in a finite number of ways. However, concepts which are produced by the finite mind may also have a fullness and depth of meaning which may approach infinity. Concepts may have meaning on many levels or horizons which may not be immediately perceived by the finite mind or imagination. Meaning, as a property belonging to concepts, may be apparent or indefinite, distinct or indistinct, clear or obscure, superficial or profound, finite in actuality or infinite in potentiality.
Vacuousness, or absence of meaning, may be negated by an infinite level of meaning. Emptiness may, or may not, be a negation of infinity. Can the infinite be empty? Can the empty be infinite? Finitude may, or may not, be a negation of infinity. Can the infinite become finite? Can the finite become infinite? These are only a few of the conceptual problems which are encountered in trying to define infinity.
1 Aristotle, Physics, Chapters 6-7, translated by R.P. Hardie and R.K. Gaye, in The Basic Works of Aristotle, ed. by Richard McKeon (Chapel Hill: Random House, 1941), pp. 328-33.
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