How do paedobaptists address the argument that infants are incapable of doing what those of Cornelius' household did?

Your argument against paedobaptism is valid if and only if (1) people have free will AND (2) there is no power as the Word of God. The heart of the objection is "how can an infant prove -- with a free will -- that it believes in Jesus Christ in order to be baptized. "

Cornelius and his household in Acts 10 are said to fear God and give alms and be pious -- things which religions that believe in human free will hold as proofs of already being saved or of at the least 'earning' salvation if such works are continued.

The whole point of Acts 10 is that although Cornelius and his household already did these things, they were NOT saved (which makes Acts 10 a chapter used to disprove "works based" salvation). They were not yet saved, and did not receive the Holy Spirit until the instant they heard the Word of God that God as Word spoke through an already saved, an already born again individual: the Apostle Peter in this instance.

Act 10:44-47 While Peter was yet speaking these words the Holy Spirit fell upon all those who were hearing the word. And the faithful of the circumcision were astonished, as many as came with Peter, that upon the nations also the gift of the Holy Spirit was poured out: for they heard them speaking with tongues and magnifying God. Then Peter answered, Can any one forbid water that these should not be baptised, who have received the Holy Spirit as we also did ?

The sequence of events: Hearing the Word of God/ being born again first, baptism second.

Other places where God directly says, as differentiated from demonstrating as in Acts 10, that we are born again by hearing the Word of God:

1 Corinthians 4:15 For if ye should have ten thousand instructors in Christ, yet not many fathers; for in Christ Jesus I have begotten you through the glad tidings.

James 1:18 Of his own will begat he us with the word of truth, that we should be a kind of firstfruits of his creatures.

1 Peter 1:22-25 Seeing ye have purified your souls in obeying the truth through the Spirit unto unfeigned love of the brethren, see that ye love one another with a pure heart fervently: 23 Being born again, not of corruptible seed, but of incorruptible, by the word of God, which liveth and abideth for ever. 24 For all flesh is as grass, and all the glory of man as the flower of grass. The grass withereth, and the flower thereof falleth away: 25 But the word of the Lord endureth for ever. And this is the word which by the gospel is preached unto you.

The whole point of ACTS 10 is that the works Cornelius was already performing were acknowledged by God and yet were insufficient for salvation, because God requires we hear Him as speech/Word of God as distinct from human language which He confounded at the attempt to build the tower at Babel:

Gen 11:6-9 And the LORD said, Behold, the people is one, and they have all one language; and this they begin to do: and now nothing will be restrained from them, which they have imagined to do.
Go to, let us go down, and there confound their language, that they may not understand one another's speech.
So the LORD scattered them abroad from thence upon the face of all the earth: and they left off to build the city.
Therefore is the name of it called Babel; because the LORD did there confound the language of all the earth: and from thence did the LORD scatter them abroad upon the face of all the earth.

God doesn't bless with a language He has Himself profaned, a language that is not-God-as-Word/Christ.

As to infant baptism and the example of Cornelius's household:
We hear the Word of God first, receive the Holy Spirit simultaneous with that hearing, are born again as a result of that hearing, recreated as new creatures in Jesus Christ, receive the Holy Spirit along with it and THEN we are baptized. Infants have nothing to prove and quite frankly, nothing to prove it with: just as unsaved adults. Most objections against infant baptism are an implication that adults have some advantage that infants don't have even though Christ says explicitly

Mark 10:15 Verily I say unto you, Whosoever shall not receive the kingdom of God as a little child, he shall not enter therein.

Infant baptism is based on the reliance of the power of the Word of God. Even in false religions that baptize children in the name of Christ, they say words in the ceremony prior to the water being splashed or "going under" in a direct attempt to mimic "Word of God first, baptism second". What baptism has anyone ever seen where no one says a word and they just use the water?

That doesn't mean that everyone who is baptized goes to heaven. False religions mimicking Christ / Word of God in human language don't work, don't birth anyone again, but they all pretend and that pretense, until God speaks to them -- as to Cornelius -- is their best honesty about reality. They are not "lying on purpose". They are pious, fear God and give alms but are not yet saved/born again. That means after they are saved, they need to be baptized for the first time the right way, not "again".

Saying that infants can't understand what is being said to them and thus can't "accept or reject" God's salvation is like saying God can't bless someone through me in English that only understands Russian. There is no such limitation in Christ. The reliance is on the power of the Word of God Himself, not human understanding. Lazarus couldn't "agree or disagree" with the Word of God that called him forth from death.

1Co 2:12-14 Now we have received, not the spirit of the world, but the spirit which is of God; that we might know the things that are freely given to us of God.
13 Which things also we speak, not in the words which man's wisdom teacheth, but which the Holy Ghost teacheth; comparing spiritual things with spiritual.
14 But the natural man receiveth not the things of the Spirit of God: for they are foolishness unto him: neither can he know them, because they are spiritually discerned.

Hope this helps.

On a controversial issue like this, we need to be careful about the approach we take and the assumptions we bring to the discussion. As a person who supports infant baptism, I will do my best to lay out my understanding of how Acts 10 is read in this issue.

I'm going to start by looking at two issues:

1. What does the text itself tell us about who was present?
2. How does our answer to the first issue relate to the other more theological elements of the question?

The text has various markers indicating who may have been present. In 10.1-2 we are introduced to Cornelius. We learn that he is head of an oikos, which is usually translated in English as either family or household. We don't know in this particular case the details of the household. But we do know that the piety of Cornelius is extended to the whole household.

In 10.3-8 Cornelius has the dream and in response he sends people to summon Peter. We learn from this section that the household includes at least two servants and a soldier/attendant (verse 7 NIV).

In 10.9-23 the focus is on Peter and his dream about the clean and unclean animals.

In 10.24-45 Peter arrives at Cornelius' home, gives his message and watches in amazement as the Spirit is poured out. Who is present? We know that Cornelius is present. We know that his relatives and close friends are present (verse 24). We don't know if his immediate household is present, but given the opening narrative and the focus on family throughout, it's a reasonable assumption.

We also know that Peter is present, and we find out in verse 45 that he has come as the leader of a group of circumcised believers. This is a detail not made clear till the final sentences of this long story. So it should alert us to the literary pattern of referring to Peter and Cornelius not just as individuals but as representatives of the two groups involved: circumcised believers and Gentiles. So we cannot assume that, at any point in the story where Peter or Cornelius alone are referred to, no one else is present.

So in summary this is a story where Cornelius and his family, household, friends and relatives are present. Peter and his fellow Jewish believers are also present. We do not and cannot know the precise make up of those two groups. All we can say is that children are quite possibly included in a household group such as this, but we don't know definitely one way or the other.

What then do we do with verses 44-46, and the statements about the Holy Spirit coming upon the Gentile group, who responded with praise and speaking in tongues. In my view a reasonable answer would be that to even ask the question is to read back into this text categories that don't apply. We have seen that the two key groups are the Jewish believers (headed by Peter) and the Gentiles (signified by Cornelius); and that we don't know the precise membership of the two groups. So the climax is not that the Spirit is poured out on adults. It's that the Spirit is poured out on Gentiles. To rule out children because children can't respond in intelligent faith is to miss the key theme of this text. It's also to make an assumption about the nature of faith which is irrelevant to the text. Why is adult faith required for those who are other than adults?

If this is a correct reading of the text, then there is another corollary. It is not a text that explicitly supports infant baptism. We can't use this text either way. We don't know conclusively if children were either present or absent. The best infant baptist view on this matter is a combination of two elements: (1) Theological - such as the idea that baptism embraces the covenantal meaning of circumcision, and is therefore applicable to the same range of people. (2) Historical - rather than look at one particular proof text such as Acts 10, the baptismal stories in the New Testament show a wide diversity of individuals (Paul and the Ethiopian eunuch), households (Cornelius and the Philippian jailer) and mass groups (Pentecost). This diversity is wide enough to embrace the faith of individual believers and the faith of parents and the church at large.