Summa Theologica, Part I (Prima Pars) From the Complete...

Obj. 3: Further, what is here in such a way as not to be elsewhere, is finite according to place. Therefore that which is a thing in such a way as not to be another thing, is finite according to substance. But God is this, and not another; for He is not a stone or wood. Therefore God is not infinite in substance.

I answer that, All the ancient philosophers attribute infinitude to the first principle, as is said (Phys. iii), and with reason; for they considered that things flow forth infinitely from the first principle. But because some erred concerning the nature of the first principle, as a consequence they erred also concerning its infinity; forasmuch as they asserted that matter was the first principle; consequently they attributed to the first principle a material infinity to the effect that some infinite body was the first principle of things.

We must consider therefore that a thing is called infinite because it is not finite. Now matter is in a way made finite by form, and the form by matter. Matter indeed is made finite by form, inasmuch as matter, before it receives its form, is in potentiality to many forms; but on receiving a form, it is terminated by that one. Again, form is made finite by matter, inasmuch as form, considered in itself, is common to many; but when received in matter, the form is determined to this one particular thing. Now matter is perfected by the form by which it is made finite; therefore infinite as attributed to matter, has the nature of something imperfect; for it is as it were formless matter. On the other hand, form is not made perfect by matter, but rather is contracted by matter; and hence the infinite, regarded on the part of the form not determined by matter, has the nature of something perfect. Now being is the most formal of all things, as appears from what is shown above (Q. 4, A. 1, Obj. 3). Since therefore the divine being is not a being received in anything, but He is His own subsistent being as was shown above (Q. 3, A. 4), it is clear that God Himself is infinite and perfect.

Reply Obj. 2: Quantity is terminated by its form, which can be seen in the fact that a figure which consists in quantity terminated, is a kind of quantitative form. Hence the infinite of quantity is the infinite of matter; such a kind of infinite cannot be attributed to God; as was said above, in this article.

Reply Obj. 3: The fact that the being of God is self-subsisting, not received in any other, and is thus called infinite, shows Him to be distinguished from all other beings, and all others to be apart from Him. Even so, were there such a thing as a self-subsisting whiteness, the very fact that it did not exist in anything else, would make it distinct from every other whiteness existing in a subject. _______________________

Objection 1: It seems that something else besides God can be essentially infinite. For the power of anything is proportioned to its essence. Now if the essence of God is infinite, His power must also be infinite. Therefore He can produce an infinite effect, since the extent of a power is known by its effect.

Obj. 2: Further, whatever has infinite power, has an infinite essence. Now the created intellect has an infinite power; for it apprehends the universal, which can extend itself to an infinitude of singular things. Therefore every created intellectual substance is infinite.

I answer that, Things other than God can be relatively infinite, but not absolutely infinite. For with regard to infinite as applied to matter, it is manifest that everything actually existing possesses a form; and thus its matter is determined by form. But because matter, considered as existing under some substantial form, remains in potentiality to many accidental forms, which is absolutely finite can be relatively infinite; as, for example, wood is finite according to its own form, but still it is relatively infinite, inasmuch as it is in potentiality to an infinite number of shapes. But if we speak of the infinite in reference to form, it is manifest that those things, the forms of which are in matter, are absolutely finite, and in no way infinite. If, however, any created forms are not received into matter, but are self-subsisting, as some think is the case with angels, these will be relatively infinite, inasmuch as such kinds of forms are not terminated, nor contracted by any matter. But because a created form thus subsisting has being, and yet is not its own being, it follows that its being is received and contracted to a determinate nature. Hence it cannot be absolutely infinite.

Reply Obj. 1: It is against the nature of a made thing for its essence to be its existence; because subsisting being is not a created being; hence it is against the nature of a made thing to be absolutely infinite. Therefore, as God, although He has infinite power, cannot make a thing to be not made (for this would imply that two contradictories are true at the same time), so likewise He cannot make anything to be absolutely infinite.

Reply Obj. 2: The fact that the power of the intellect extends itself in a way to infinite things, is because the intellect is a form not in matter, but either wholly separated from matter, as is the angelic substance, or at least an intellectual power, which is not the act of any organ, in the intellectual soul joined to a body.

Reply Obj. 3: Primary matter does not exist by itself in nature, since it is not actually being, but potentially only; hence it is something concreated rather than created. Nevertheless, primary matter even as a potentiality is not absolutely infinite, but relatively, because its potentiality extends only to natural forms. _______________________

Objection 1: It seems that there can be something actually infinite in magnitude. For in mathematics there is no error, since "there is no lie in things abstract," as the Philosopher says (Phys. ii). But mathematics uses the infinite in magnitude; thus, the geometrician in his demonstrations says, "Let this line be infinite." Therefore it is not impossible for a thing to be infinite in magnitude.

Obj. 2: Further, what is not against the nature of anything, can agree with it. Now to be infinite is not against the nature of magnitude; but rather both the finite and the infinite seem to be properties of quantity. Therefore it is not impossible for some magnitude to be infinite.

Obj. 3: Further, magnitude is infinitely divisible, for the continuous is defined that which is infinitely divisible, as is clear from Phys. iii. But contraries are concerned about one and the same thing. Since therefore addition is opposed to division, and increase opposed to diminution, it appears that magnitude can be increased to infinity. Therefore it is possible for magnitude to be infinite.

Obj. 4: Further, movement and time have quantity and continuity derived from the magnitude over which movement passes, as is said in Phys. iv. But it is not against the nature of time and movement to be infinite, since every determinate indivisible in time and circular movement is both a beginning and an end. Therefore neither is it against the nature of magnitude to be infinite.

On the contrary, Every body has a surface. But every body which has a surface is finite; because surface is the term of a finite body. Therefore all bodies are finite. The same applies both to surface and to a line. Therefore nothing is infinite in magnitude.

I answer that, It is one thing to be infinite in essence, and another to be infinite in magnitude. For granted that a body exists infinite in magnitude, as fire or air, yet this could not be infinite in essence, because its essence would be terminated in a species by its form, and confined to individuality by matter. And so assuming from these premises that no creature is infinite in essence, it still remains to inquire whether any creature can be infinite in magnitude.

Now it is manifest that a natural body cannot be actually infinite. For every natural body has some determined substantial form. Since therefore the accidents follow upon the substantial form, it is necessary that determinate accidents should follow upon a determinate form; and among these accidents is quantity. So every natural body has a greater or smaller determinate quantity. Hence it is impossible for a natural body to be infinite. The same appears from movement; because every natural body has some natural movement; whereas an infinite body could not have any natural movement; neither direct, because nothing moves naturally by a direct movement unless it is out of its place; and this could not happen to an infinite body, for it would occupy every place, and thus every place would be indifferently its own place. Neither could it move circularly; forasmuch as circular motion requires that one part of the body is necessarily transferred to a place occupied by another part, and this could not happen as regards an infinite circular body: for if two lines be drawn from the centre, the farther they extend from the centre, the farther they are from each other; therefore, if a body were infinite, the lines would be infinitely distant from each other; and thus one could never occupy the place belonging to any other.

The same applies to a mathematical body. For if we imagine a mathematical body actually existing, we must imagine it under some form, because nothing is actual except by its form; hence, since the form of quantity as such is figure, such a body must have some figure, and so would be finite; for figure is confined by a term or boundary.