When were these Gentiles appointed to aeonous life?

This is a discussion on the meaning of Act 13:48
"And when the Gentiles heard this, they were glad, and glorified the word of the Lord: and as many as were ordained to eternal life believed."

Questions one should ask are: "What does the context indicate that they heard that caused them to rejoice and believe? What was it that the context indicates they believed? Who does the context indicate it was that believed? For instance, in John 20, John says that when he and Peter saw the linen cloths John believed, but immediately in the next verse John admits that neither he nor Peter yet understood that Jesus must rise from the dead.
Jhn 20:8 "Then went in also that other disciple, which came first to the sepulchre, and he saw, and believed.
Jhn 20:9 For as yet they knew not the scripture, that he must rise again from the dead."

So, what did John believe? He believed Mary's assessment: "They have taken the body".
Jhn 20:2 Then she runneth, and cometh to Simon Peter, and to the other disciple, whom Jesus loved, and saith unto them, They have taken away the Lord out of the sepulchre, and we know not where they have laid him.

Likewise in Acts 13. What the audience heard that cause them to rejoice was the following -
Act 13:46 - 48 Then Paul and Barnabas waxed bold, and said, It was necessary that the word of God should first have been spoken to you: but seeing ye put it from you, and judge yourselves unworthy of everlasting life, lo, we turn to the Gentiles. For so hath the Lord commanded us, saying, I have set thee to be a light of the Gentiles, that thou shouldest be for salvation unto the ends of the earth." And when the Gentiles heard THIS..... they were glad, and glorified the word of the Lord: and as many as were ordained to eternal life believed (THIS).

So, they believed THIS after having heard Paul's preaching of the gospel during which some had heard the gospel and had been appointed/had arranged themselves to aeonous life. We are ordained to aeonous life by believing the gospel which the jews had been contradicting.

Act 13:45 But when the Jews saw the multitudes, they were filled with envy, and spake against those things which were spoken by Paul, contradicting and blaspheming

Also there were Gentiles there who has heard Paul a week earlier and had believed the gospel, or had at least been open to it, because Paul had encouraged them to continue in the grace of God. So some of Paul's audience may have been appointed/arranged themselves for aeonous life during the previous week.
Act 13:43 Now when the congregation was broken up, many of the Jews and religious proselytes followed Paul and Barnabas: who, speaking to them, persuaded them to continue in the grace of God.

So, the context indicates to me that those who had listened open-heartedly to Paul the previous week and during Paul's present preaching had been appointed/had arranged themselves to aeonous life by believing the gospel, after which they heard Paul say He was being sent by God to the Gentiles, news that they believed and rejoiced to hear.

There is no indication from the text that this appointing had been done any earlier than the previous sabbath.

The specific message was that the Gospel (which the Gentiles had previously heard) was for the Gentiles as much as for the Jews and indeed, "salvation unto the ends of the earth" was ordained. That is what they had rejoiced to hear.

So, the Gentiles who had heard the gospel, here hear that it is for them, and believing it, rejoiced, not that they didn't believe it when they first had heard it, they just didn't know for sure that it was meant for them. Wouldn't this be at least one aspect of 'election' that the Jews would have a very strong objection to?

Also there were Gentiles there who has heard Paul a week earlier and had believed the gospel, or had at least been open to it, because Paul had encouraged them to continue in the grace of God. So some of Paul's audience may have been appointed/arranged themselves for aeonous life during the previous week.

The specific message was that the Gospel (which the Gentiles had previously heard) was for the Gentiles as much as for the Jews and indeed, "salvation unto the ends of the earth" was ordained. That is what they had rejoiced to hear.

So, the Gentiles who had heard the gospel, here hear that it is for them, and believing it, rejoiced, not that they didn't believe it when they first had heard it, they just didn't know for sure that it was meant for them. Wouldn't this be at least one aspect of 'election' that the Jews would have a very strong objection to?

Jews and gentiles had heard the gospel the previous week.

Beginning in Acts 13:16, Paul went through the history of the Jews ... probably for the benefit of the gentiles present ... and then we read concerning the Lord Jesus Christ:

Acts 13:

37 But he, whom God raised again, saw no corruption.

38 Be it known unto you therefore, men and brethren, that through this man is preached unto you the forgiveness of sins:

39 And by him all that believe are justified from all things, from which ye could not be justified by the law of Moses.

40 Beware therefore, lest that come upon you, which is spoken of in the prophets;

41 Behold, ye despisers, and wonder, and perish: for I work a work in your days, a work which ye shall in no wise believe, though a man declare it unto you.

42 And when the Jews were gone out of the synagogue, the Gentiles besought that these words might be preached to them the next sabbath.

The gentiles (and hopefully the Jews) went out and told others about the preaching of Paul. They brought their friends and families back the next week ...

vs 44 - almost the whole city came to hear the Word of God.

vs 45 - Jews saw the multitudes and were filled with envy, and spake against those things which were spoken by Paul, contradicting and blaspheming

vs 46-47 - Paul and Barnabas didn't shrink away ... they told the Jews It was necessary that the word of God should first have been spoken to you: but seeing ye put it from you, and judge yourselves unworthy of everlasting life, lo, we turn to the Gentiles. For so hath the Lord commanded us, saying, I have set thee to be a light of the Gentiles, that thou shouldest be for salvation unto the ends of the earth.

vs 48 - the Gentiles had always been treated as dogs by the Jews. But Paul and Barnabas had preached the truth to them ... And when the Gentiles heard this, they were glad, and glorified the word of the Lord: and as many as were ordained to eternal life believed.

The gentiles learned that they did not have to convert to Judaism and that God had opened the floodgates to non-Jews. That's why they were glad, and glorified the Word of Lord.

I love that the folks who heard Paul the week before had gone out, grabbed their friends and families, and brought them back to hear Paul the next week.

At the church I attend, there's a sign as we all leave the parking lot ... "You are now entering the mission field". I love that sign. We are filled with God's Word at church and we go out into the mission field and speak the truth of God's Word to others. God has always used believers to reach unbelievers.
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In the Bible, "aeonian life," translated from the Greek word "aionios," refers to the life of the coming age, often associated with the resurrection and a new, transformed existence. It's not just indefinite duration, but a quality of life distinct from the present age, emphasizing a relationship with God and the messianic kingdom.