FOR THE PROMISE TO ABRAHAM OR TO HIS DESCENDANTS : ou gar dia nomou (Not thru law = first for emphasis) e epaggelia to abraam e to spermati autou: (Ge 12:3; 17:4,5,16; 22:17,18; 28:14; 49:10; Ps 2:8; 72:11)

For (gar) Paul begins with this conjunction (gar) which serves as a marker of cause or reason between events, to amplify why it is that those who "are of the circumcision" (v12) can only inherit the "promise of Abraham" "through the righteousness of faith" and "not through the Law" or as he describes it in (v12) by following "in the steps of the faith of" their "father Abraham". “

The argument continues relentlessly on as Paul chases every possible objector down every possible alleyway of logic and Scripture.” The apostle now must deal with the objection that blessing came through the law and that therefore the Gentiles who did not know the law were cursed (see Jn 7:49).

Note that Paul in the Greek begins with the phrase "for not thru the law" which shows what he desires to emphasize that righteousness is not via the Law.

Promise (1860) (epaggelia form epí  = intensifier or upon + aggéllo = tell, declare) means an announcement upon and was primarily legal term denoting summons or a promise to do or give something. It is used only of the promises of God (except Acts 23:21) and refers to a thing promised, a gift graciously given and is not a pledge secured by negotiation.

Many Jews thought that the privileges they enjoyed came from their adherence to the Mosaic Law. This is false. The promises were given to Abraham 430 years before the law was given; hence the law does not annul the Abrahamic covenant. It was merely added alongside until Christ should come to fulfill it (Gal 3:17-19).

THAT HE WOULD BE HEIR OF THE WORLD : to kleronomon auton einai (PAN) tou kosmou:

Heir (2818) (kleronomos from kleros = lot + nomos = something parceled out, allotted) means a sharer by lot, an inheritor, a possessor. In the Greco-Roman world the kleronomos was a legal term.

Kleronomos signifies more than one who inherits, or obtains a portion, for it means to take into possession. Kleronomos while being virtually a title, also conveys the significance of dominion and authority.

What does Paul mean "heir of the world"? The context is those who by faith are the spiritual descendants of Abraham, so this might refer to a "world w/o end", that which is to come when Christ sets up His kingdom on earth and then when a new Jerusalem, new Heaven and new Earth are brought into existence by God after world is burned with fire. So Abraham's SPIRITUAL heirs receive a "KINGDOM WHICH CANNOT BE SHAKEN", a better country, a city God is preparing (Heb 11:13-15,16, Jn 14:3, etc).

Expositor's Bible Commentary however says

"The thought moves on to the consideration that Abraham’s justification was apart from the law or legal considerations. Paul speaks of a promise received by “Abraham and his offspring” that “he should be heir of the world.” Nothing so precise can be detected in the text of Genesis. Meyer suggests that the possession of the land of promise, in accordance with God’s gift, is here looked on as the foil for a greater inheritance, namely, the messianic kingdom, in which the descendants of Abraham would have a special stake. The objection to this is certainly not that no place for a messianic kingdom can be found in Paul (Ro 8:17; 1Co 6:2; cf. Mt 19:28), but rather that the subject of the chapter has not changed. Abraham’s justification came in connection with the promise of offspring comparable in number to the stars of the heavens. Nothing in the section we are considering suggests the thought of dominion. To be sure, Abraham received a promise that his descendants would possess the gate of their enemies (Ge 22:17) but that concept is not introduced here. Furthermore, “world” lacks the article, so that it is not likely intended to denote the physical world but the multitude of those who will follow Abraham in future generations in terms of his faith. These he can claim as his own. Finally, it is not said that Abraham’s offspring will be heirs of the world but that he will be such an heir. This is not favorable to the eschatological interpretation of a millennial kingdom involving the renewed nation of Israel as its core element. The theme is still that vast influence of the man of faith upon succeeding generations and peoples. He will be the father of many nations in the sense that he is the father of their faith, since by that means rather than by some other they will be justified (Ge12:3; 22:18)." (Gaebelein, F, Editor: Expositor's Bible Commentary 6-Volume New Testament. Zondervan Publishing)

This refers to Christ and is the essence of the covenant God made with Abraham and his descendants. The final provision of that covenant was that through Abraham’s seed all the world would be blessed (Ge12:3). Paul argues that “the seed” refers specifically to Christ and that this promise really constituted the gospel (Gal 3:8, 16; Jn 8:56). All believers, by being in Christ, become heirs of the promise (Gal 3:29; 1Cor 3:21-23).

Keep in mind that Abraham’s PHYSICAL descendants include not only the Jewish people, but also the Arab world (thru Ishmael) and the nations listed in [Ge 25:1-4]. All who trust Jesus Christ as Savior are SPIRITUAL children of Abraham (Gal 3:6-9), and that will be a vast multitude (Rev 7:9).

Piper explains that

"eternity (he's talking about being either an heir of the world or an heir of wrath) compares to this life the way the Rocky Mountains compare to the ripples of an orange peel." (Inheriting the World Depends on Faith, Not Law)

Vincent writes that

“Paul here takes the Jewish conception of the universal dominion of the Messianic theocracy prefigured by the inheritance of Canaan, divests it of its Judaistic element, and raises it to a Christological truth.” (Vincent's comment is a bit difficult to follow but I think is reasonably accurate assessment of what it means to be heir of the world). (Romans 4: Greek Word Studies)

World (2889) (kosmos from komeo = tend, take care of) can refer to an orderly arrangement.  Kosmos speaks of an ordered system and gives us English cosmetic which has the basic meaning of covering up disorder with something that brings order.

The Greeks had a word, chaos which comes into our language in its exact spelling, “chaos” and was used by pagan Greek philosophers of what they considered to be the first state of the universe. The word meant “unformed matter.” It spoke of darkness, a vast gulf or chasm, a pit, the nether abyss. But the Bible writers speak of the original state of the universe as one of a harmonious arrangement of things, a kosmos not a chaos. God, speaking of the laying of the cornerstone of the universe, speaks of the sons of God, the angels, shouting for joy at its creation (Job 38:6, 7). The holy angels did not shout for joy over a chaos.

WAS NOT THROUGH (obedience to) THE LAW BUT THROUGH THE RIGHTEOUSNESS OF FAITH : ou gar dia nomou (Not thru law = first for emphasis) alla dia dikaiosunes pisteos: (Gal 3:16-18,29)

Paul explains that it could not have been through the "Law" for the "Law" had not even been given at the time God cut His covenant of promise with Abraham...

Now the promises were spoken to Abraham and to his seed. He does not say, "And to seeds," as referring to many, but rather to one, "And to your seed," that is, Christ. What I am saying is this: the Law, which came four hundred and thirty years later, does not invalidate a covenant previously ratified by God (Genesis 12:1-3, 15:18), so as to nullify (make void) the promise (to Abraham that in him "all the families of the earth shall be blessed." ). For if the inheritance is based on law, it is no longer based on a promise; but God has granted it to Abraham by means of a promise. (Gal 3:16-18)

29 And if you belong to Christ, then you are Abraham's offspring, heirs according to promise. (Gal 3:29)

The fulfillment of the promise was not as a result of Abraham’s keeping the law. Just as Abraham was not justified by the rite of circumcision (Ro 4:9-12), neither was he justified by keeping the Mosaic law (Ro 4:13-15) which did not even exist when the promise was made to Abraham.

Righteousness received from God by faith (see note Romans 1:17).

Righteousness (1343) (dikaiosune) is rightness of character before God and rightness of actions before men. Both of these qualities are based on truth, which is conformity to the Word and will of God. Righteousness is attitude and action which conforms to a standard and can be either man's imperfect standard (as exemplified by the self-righteous Pharisees) or God's standard of perfect holiness.

Wuest comments:

FOR IF THOSE WHO ARE OF THE LAW ARE HEIRS: ei gar oi ek nomou kleronomoi: (16; Gal 2:21; 3:18-24; 5:4; Phil 3:9; Heb 7:19,28)

Law (3551) (nomos from némo = divide among, parcel out, allot) is etymologically something parceled out, allotted, what one has in use and possession; hence, usage, custom. In the present context "nomos" refers to the Mosaic system of legislation. The law became so exalted by the rabbis in Judaism that it became the explanation and justification of Israel's existence. However, at the heart of OT religion was the covenant and not the law, which was only a standard of obedience necessary for the preservation of the covenant relationship. In postexilic Israel, obedience to the law was the necessary condition to become a member of God's people and thus the law became more central than covenant in the religion of Judaism.

If only those who perfectly keep the law—an impossibility—receive the promise, faith has no value.

If people could receive what God promised by following the law, then faith is worthless. And God's promise to Abraham is worthless.

A. T. Robertson writes that

If legalists are heirs of the Messianic promise to Abraham (condition of first class, assumed as true for argument’s sake), the faith is emptied of all meaning...and the promise to Abraham is made permanently idle

Judaizers wanted their hearers to go back to Moses, but that was not far enough. We must go back to Abraham where the promise started. The Law did not annul the promise. The Law was given to reveal sin (see note Romans 3:20) and prepare the way for Christ to come and fulfill the promise (Gal 3:24,25). The law is a tutor, not a savior; a mirror, not a cleanser. (For summary see Purpose of the Law)

FAITH IS MADE VOID AND THE PROMISE IS NULLIFIED: kekenotai (3SRPI) e pistis kai katergetai (3SRPI) e epaggelia: (Ro 3:31; Nu 30:12,15; Ps 119:126; Isa 55:11; Jer 19:7)

Made Void (2758) (kenoo from kenos = empty, void) means deprived of its effect or divested of its power or force. The idea is to take away the power of something and so to make it of no meaning or effect. It meant to render vain or useless.

Faith is made futile and empty of all meaning because if accessing of the promise was through keeping the Law then it would not be based on trust but on what I could do.

William MacDonald explains "faith is made void" writing...

"Faith is set aside because it is a principle that is completely opposite to law: faith is a matter of believing, while law is a matter of doing. The promise would then be worthless because it would be based on conditions that no one would be able to meet." (MacDonald, W., & Farstad, A. Believer's Bible Commentary: Old and New Testaments. Nashville: Thomas Nelson)

Nullified (2673) (katargeo from kata = intensifies meaning + argeo = to be idle or inactive from argos = ineffective, idle, inactive from  a = without + érgon = work) literally means to reduce to inactivity. The idea is to make the power or force of something ineffective and so to render powerless, reduce to inactivity. To do away with. To put out of use.  To cause to be idle or useless. To render entirely idle, inoperative or ineffective. Cause something to come to an end or cause it to cease to happen. To abolish or cause not to function. To free or release from an earlier obligation or relationship. To no longer take place.

Katargeo is in the perfect tense which speaks of a permanent effect. That is if those who were under the Law were those who would become the legitimate heirs of the promises made to Abraham, then this would permanently nullify faith or make faith forever of no effect. Praise God, that is not the case as Paul explains!

Vine explains that katargeo...

never means “to annihilate.” (= to destroy utterly and completely and thus cause to cease to exist) The general idea in the word is that of depriving a thing of the use for which it is intended. Thus it implies, not loss of being, but loss of well-being (Ed note: although this latter idea cannot be easily applied to many the NT occurrences which refer to inanimate things such as the Law, death, the power of sin, etc). (Vine, W. Collected writings of W. E. Vine. Nashville: Thomas Nelson )

Katargeo is clearly a "Pauline verb" as shown by the 27 uses (15 in the passive voice) in the NASB (Lu; 6x Romans; 8x 1Co; 4x 2Co; 3x Gal; Ep; 2Th; 2Ti; Heb) and is translated: abolished, 4; abolishing, 1; bring to an end, 1; did away, 1; do away, 1; done away, 4; fades away, 1; fading, 1; fading away, 1; nullified, 1; nullify, 4; passing away, 1; released, 2; removed, 1; render powerless, 1; severed, 1; use, 1.

Consider lingering over and pondering all the NT uses of katargeo (click the links above or study through many of the NT uses noted below) to mine the encouraging, edifying truths associated with this great "Pauline verb." You will not be disappointed.

Comment: The fact that many people reject the Scriptures is of no effect = katargeo, for as the Psalmist testifies "Forever, O LORD, Thy word is settled in heaven." -- Psalm 119:89.  God remains faithful to His covenant promises to Israel originally given to Abraham -- beginning in Genesis 12:1-3 -- and passed on to the lines of Isaac and Jacob. The OT amply testifies to God's faithfulness in the face of Israel's unfaithfulness. For example, of the generation that received the law at Mt Sinai, only two adults proved faithful, Caleb and Joshua. Nevertheless, God brought the whole nation of Israel into the land of Canaan as He had promised, though the unbelieving generation died in the wilderness.) In short, to insist that the lack of faith of those who were entrusted with God's Word hardly makes that Word of no effect (= katargeo).

Comment: In other words Paul is saying that if one emphasizes the vital role of faith in salvation, by do doing he is not making the Law of no use = katargeo.  Click here to read a summary of the 4 effects of the Law.) Paul's point is that justification by faith does not invalidate the purpose of the Law but in fact establishes it because Law was not meant to save to reveal sin.

Romans 6:6 knowing this, that our old self (see discussion of "old man") was crucified with Him, that our body of sin might be done away with (katargeo = make ineffective by removing its power of control) that we should no longer be slaves to sin.

Comment: Yes, Sin as a power, is made inoperative, but it is not annihilated as all believers have experienced. When we as believers make "wrong choices" (or let's call it what it is, when we sin, doing our own thing, going our own way) we can still ''recharge'' or "revive" the old master ("the Sin") that persists latent within every believer. 

Before Christ gave us a new heart, "the Sin" within us ruled us, wielding a power over us which we could not resist; and which led us to commit sins. The law functioned to arouse the sinful desire, but no longer has that effect (unless we choose to put ourselves back up under a list of do's and don'ts). But now that our old self has been nailed to the cross of Christ, the power of sin and the effect of the Law over our physical bodies have been rendered inoperative (katargeo). (for more discussion see note Romans 6:6)

Comment: So we see that the "law" that Paul refers to, that they would be familiar with and which says a woman is bound to her husband, is no longer in effect if the husband dies. The hold it once had on the wife is now nullified. Look at verse 3: "So then if, while her husband is living, she is joined to another man, she shall be called an adulteress; but if her husband dies, she is free from the law, so that she is not an adulteress, though she is joined to another man." To reiterate, these verses teach that as death releases a widow from the law of marriage, so Christ's death releases believers who are united with him from the Law. Constable adds that "It is as though God shifted the transmissions of our lives into neutral gear. Now something else drives our lives, namely the Holy Spirit. Sin and the Law no longer drive us forward, though we can engage those powers if we choose to do so and take back control of our lives from God." ( Tom Constable's Expository Notes)